the laplace transform of the dirac delta function · de & ivp bernd schroder¨ louisiana tech...
TRANSCRIPT
![Page 1: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/1.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
The Laplace Transform of The DiracDelta Function
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 2: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/2.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It Was
No matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 3: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/3.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 4: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/4.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 5: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/5.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
OriginalDE & IVP
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 6: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/6.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
OriginalDE & IVP
-L
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 7: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/7.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
-L
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 8: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/8.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
-L
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 9: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/9.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
-L
Algebraic solution,partial fractions
?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 10: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/10.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
Laplace transformof the solution
-L
Algebraic solution,partial fractions
?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 11: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/11.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
Laplace transformof the solution
-
L
L −1
Algebraic solution,partial fractions
?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 12: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/12.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Everything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace transforms stays the same.
Time Domain (t) Transform domain (s)
OriginalDE & IVP
Algebraic equation forthe Laplace transform
Laplace transformof the solutionSolution
-
L
L −1
Algebraic solution,partial fractions
?
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 13: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/13.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What is the Delta Function?
1. δ (x) = 0 for all x 6= 0.
2. Sifting property:∫
∞
−∞
f (x)δ (x−a) dx = f (a) ????
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 14: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/14.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What is the Delta Function?1. δ (x) = 0 for all x 6= 0.
2. Sifting property:∫
∞
−∞
f (x)δ (x−a) dx = f (a) ????
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 15: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/15.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What is the Delta Function?1. δ (x) = 0 for all x 6= 0.
2. Sifting property:∫
∞
−∞
f (x)δ (x−a) dx = f (a)
????
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 16: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/16.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What is the Delta Function?1. δ (x) = 0 for all x 6= 0.
2. Sifting property:∫
∞
−∞
f (x)δ (x−a) dx = f (a) ????
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 17: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/17.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What is the Delta Function?1. δ (x) = 0 for all x 6= 0.
2. Sifting property:∫
∞
−∞
f (x)δ (x−a) dx = f (a)
3. The delta function is used to model “instantaneous” energytransfers.
4. L
δ (t−a)
= e−as
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 18: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/18.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What is the Delta Function?1. δ (x) = 0 for all x 6= 0.
2. Sifting property:∫
∞
−∞
f (x)δ (x−a) dx = f (a)
3. The delta function is used to model “instantaneous” energytransfers.
4. L
δ (t−a)
= e−as
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 19: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/19.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What is the Delta Function?1. δ (x) = 0 for all x 6= 0.
2. Sifting property:∫
∞
−∞
f (x)δ (x−a) dx = f (a)
3. The delta function is used to model “instantaneous” energytransfers.
4. L
δ (t−a)
= e−as
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 20: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/20.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application
(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′+8q′+15q = 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 21: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/21.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′+8q′+15q = 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 22: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/22.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.
Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′+8q′+15q = 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 23: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/23.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′+8q′+15q = 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 24: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/24.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′+8q′+15q = 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 25: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/25.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′
+8q′+15q = 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 26: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/26.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′+8q′
+15q = 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 27: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/27.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′+8q′+15q
= 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 28: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/28.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
A Possible Application(Dimensions are fictitious.)
In an LRC circuit with L = 1H, R = 8Ω and C =115
F, thecapacitor initially carries a charge of 1C and no currents areflowing. There is no external voltage source. At time t = 2s, apower surge instantaneously applies an impulse of 4δ (t−2)into the system.Describe the charge of the capacitor over time.
Lq′′+Rq′+qC
= E(t)
q′′+8q′+15q = 4δ (t−2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 29: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/29.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s+8sQ−8+15Q = 4e−2s(
s2 +8s+15)
Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 30: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/30.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)
s2Q− s+8sQ−8+15Q = 4e−2s(s2 +8s+15
)Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 31: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/31.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s
+8sQ−8+15Q = 4e−2s(s2 +8s+15
)Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 32: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/32.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s+8sQ−8
+15Q = 4e−2s(s2 +8s+15
)Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 33: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/33.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s+8sQ−8+15Q
= 4e−2s(s2 +8s+15
)Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 34: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/34.jpg)
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Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s+8sQ−8+15Q = 4e−2s
(s2 +8s+15
)Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 35: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/35.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s+8sQ−8+15Q = 4e−2s(
s2 +8s+15)
Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 36: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/36.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s+8sQ−8+15Q = 4e−2s(
s2 +8s+15)
Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 37: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/37.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s+8sQ−8+15Q = 4e−2s(
s2 +8s+15)
Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 38: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/38.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
q′′+8q′+15q = 4δ (t−2)s2Q− s+8sQ−8+15Q = 4e−2s(
s2 +8s+15)
Q = s+8+4e−2s
(keep the exponential separate)
Q =s+8
s2 +8s+15+ e−2s 4
s2 +8s+15
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 39: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/39.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 40: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/40.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.
s+8(s+3)(s+5)
=A
s+3+
Bs+5
s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 41: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/41.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5
s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 42: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/42.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 43: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/43.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 :
5 = 2A, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 44: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/44.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A
, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 45: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/45.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 46: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/46.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5
3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 47: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/47.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5 3 =−2B
, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 48: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/48.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 49: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/49.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.s+8
(s+3)(s+5)=
As+3
+B
s+5s+8 = A(s+5)+B(s+3)
s =−3 : 5 = 2A, A =52
s =−5 3 =−2B, B =−32
s+8(s+3)(s+5)
=52
1s+3
− 32
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 50: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/50.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 : 4 = 2A, A = 2s =−5 : 4 =−2B, B =−2
4(s+3)(s+5)
= 21
s+3−2
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 51: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/51.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.
4(s+3)(s+5)
=A
s+3+
Bs+5
4 = A(s+5)+B(s+3)s =−3 : 4 = 2A, A = 2s =−5 : 4 =−2B, B =−2
4(s+3)(s+5)
= 21
s+3−2
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 52: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/52.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+5
4 = A(s+5)+B(s+3)s =−3 : 4 = 2A, A = 2s =−5 : 4 =−2B, B =−2
4(s+3)(s+5)
= 21
s+3−2
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 53: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/53.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 : 4 = 2A, A = 2s =−5 : 4 =−2B, B =−2
4(s+3)(s+5)
= 21
s+3−2
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 54: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/54.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 :
4 = 2A, A = 2s =−5 : 4 =−2B, B =−2
4(s+3)(s+5)
= 21
s+3−2
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 55: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/55.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 : 4 = 2A
, A = 2s =−5 : 4 =−2B, B =−2
4(s+3)(s+5)
= 21
s+3−2
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 56: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/56.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 : 4 = 2A, A = 2
s =−5 : 4 =−2B, B =−24
(s+3)(s+5)= 2
1s+3
−21
s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 57: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/57.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 : 4 = 2A, A = 2s =−5 :
4 =−2B, B =−24
(s+3)(s+5)= 2
1s+3
−21
s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 58: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/58.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 : 4 = 2A, A = 2s =−5 : 4 =−2B
, B =−24
(s+3)(s+5)= 2
1s+3
−21
s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 59: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/59.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 : 4 = 2A, A = 2s =−5 : 4 =−2B, B =−2
4(s+3)(s+5)
= 21
s+3−2
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 60: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/60.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Partial fraction decompositions.4
(s+3)(s+5)=
As+3
+B
s+54 = A(s+5)+B(s+3)
s =−3 : 4 = 2A, A = 2s =−5 : 4 =−2B, B =−2
4(s+3)(s+5)
= 21
s+3−2
1s+5
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 61: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/61.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t− 32
e−5t +U (t−2)[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 62: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/62.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t− 32
e−5t +U (t−2)[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 63: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/63.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]
q =52
e−3t− 32
e−5t +U (t−2)[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 64: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/64.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t− 32
e−5t +U (t−2)[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 65: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/65.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t
− 32
e−5t +U (t−2)[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 66: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/66.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t− 32
e−5t
+U (t−2)[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 67: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/67.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t− 32
e−5t +U (t−2)
[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 68: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/68.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t− 32
e−5t +U (t−2)[2e−3t−2e−5t
]
t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 69: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/69.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t− 32
e−5t +U (t−2)[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 70: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/70.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Solve the Initial Value Problemq′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0
Q =s+8
(s+3)(s+5)+ e−2s 4
(s+3)(s+5)
Q =52
1s+3
− 32
1s+5
+ e−2s[
21
s+3−2
1s+5
]q =
52
e−3t− 32
e−5t +U (t−2)[2e−3t−2e−5t
]t→t−2
=52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 71: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/71.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What Happens in the Physical System?
q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 72: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/72.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What Happens in the Physical System?
q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 73: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/73.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What Happens in the Physical System?
q′ =−152
e−3t +152
e−5t +U (t−2)[−6e−3(t−2) +10e−5(t−2)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 74: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/74.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
What Happens in the Physical System?
q′ =−152
e−3t +152
e−5t +U (t−2)[−6e−3(t−2) +10e−5(t−2)
]Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 75: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/75.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 76: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/76.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 77: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/77.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 78: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/78.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 79: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/79.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 80: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/80.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 81: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/81.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0
√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 82: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/82.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 83: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/83.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0)
=52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 84: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/84.jpg)
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Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0
= 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 85: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/85.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1
√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 86: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/86.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 87: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/87.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0)
= −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 88: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/88.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 89: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/89.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 90: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/90.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial
Value Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
First consider q1 =52
e−3t− 32
e−5t.
15(
52
e−3t− 32
e−5t)
+8(−15
2e−3t +
152
e−5t)
+(
452
e−3t− 752
e−5t)
=(
752− 120
2+
452
)e−3t +
(−45
2+
1202− 75
2
)e−5t
= 0√
q1(0) =52
e−3·0− 32
e−5·0 = 1√
q′1(0) = −152
e−3·0 +152
e−5·0 = 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 91: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/91.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial Value
Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
Now consider q2 = 2e−3(t−2)−2e−5(t−2).
15(2e−3(t−2)−2e−5(t−2)
)+8
(−6e−3(t−2)+10e−5(t−2)
)+
(18e−3(t−2)−50e−5(t−2)
)= (30−48+18)e−3(t−2) +(−30+80−50)e−5(t−2)
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 92: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/92.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial Value
Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
Now consider q2 = 2e−3(t−2)−2e−5(t−2).
15(2e−3(t−2)−2e−5(t−2)
)+8
(−6e−3(t−2)+10e−5(t−2)
)+
(18e−3(t−2)−50e−5(t−2)
)= (30−48+18)e−3(t−2) +(−30+80−50)e−5(t−2)
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 93: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/93.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial Value
Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
Now consider q2 = 2e−3(t−2)−2e−5(t−2).
15(2e−3(t−2)−2e−5(t−2)
)
+8(−6e−3(t−2)+10e−5(t−2)
)+
(18e−3(t−2)−50e−5(t−2)
)= (30−48+18)e−3(t−2) +(−30+80−50)e−5(t−2)
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 94: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/94.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial Value
Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
Now consider q2 = 2e−3(t−2)−2e−5(t−2).
15(2e−3(t−2)−2e−5(t−2)
)+8
(−6e−3(t−2)+10e−5(t−2)
)
+(18e−3(t−2)−50e−5(t−2)
)= (30−48+18)e−3(t−2) +(−30+80−50)e−5(t−2)
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 95: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/95.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial Value
Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
Now consider q2 = 2e−3(t−2)−2e−5(t−2).
15(2e−3(t−2)−2e−5(t−2)
)+8
(−6e−3(t−2)+10e−5(t−2)
)+
(18e−3(t−2)−50e−5(t−2)
)
= (30−48+18)e−3(t−2) +(−30+80−50)e−5(t−2)
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 96: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/96.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial Value
Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
Now consider q2 = 2e−3(t−2)−2e−5(t−2).
15(2e−3(t−2)−2e−5(t−2)
)+8
(−6e−3(t−2)+10e−5(t−2)
)+
(18e−3(t−2)−50e−5(t−2)
)= (30−48+18)e−3(t−2) +(−30+80−50)e−5(t−2)
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 97: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/97.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial Value
Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
Now consider q2 = 2e−3(t−2)−2e−5(t−2).
15(2e−3(t−2)−2e−5(t−2)
)+8
(−6e−3(t−2)+10e−5(t−2)
)+
(18e−3(t−2)−50e−5(t−2)
)= (30−48+18)e−3(t−2) +(−30+80−50)e−5(t−2)
= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function
![Page 98: The Laplace Transform of The Dirac Delta Function · DE & IVP Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Laplace Transform of The Dirac Delta](https://reader031.vdocument.in/reader031/viewer/2022022718/5c5c72f209d3f2e54d8b4da3/html5/thumbnails/98.jpg)
logo1
Transforms and New Formulas A Model The Initial Value Problem Interpretation Double Check
Does q =52
e−3t− 32
e−5t +U (t−2)[2e−3(t−2)−2e−5(t−2)
]Solve the Initial Value
Problem q′′+8q′+15q = 4δ (t−2), q(0) = 1, q′(0) = 0?
Now consider q2 = 2e−3(t−2)−2e−5(t−2).
15(2e−3(t−2)−2e−5(t−2)
)+8
(−6e−3(t−2)+10e−5(t−2)
)+
(18e−3(t−2)−50e−5(t−2)
)= (30−48+18)e−3(t−2) +(−30+80−50)e−5(t−2)
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
The Laplace Transform of The Dirac Delta Function