laplace transforms of periodic functions · 2008. 11. 6. · periodic functions 1. a function f is...
TRANSCRIPT
![Page 1: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/1.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Laplace Transforms of Periodic Functions
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 2: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/2.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 3: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/3.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 4: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/4.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 5: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/5.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 6: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/6.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 7: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/7.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 8: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/8.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 9: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/9.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 10: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/10.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 11: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/11.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 12: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/12.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
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
Laplace Transforms of Periodic Functions
![Page 13: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/13.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Periodic Functions
1. A function f is periodic with period T > 0 if and only if forall t we have f (t +T) = f (t).
2. If f is bounded, piecewise continuous and periodic withperiod T , then
L{
f (t)}
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 14: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/14.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Periodic Functions1. A function f is periodic with period T > 0 if and only if for
all t we have f (t +T) = f (t).
2. If f is bounded, piecewise continuous and periodic withperiod T , then
L{
f (t)}
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 15: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/15.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Periodic Functions1. A function f is periodic with period T > 0 if and only if for
all t we have f (t +T) = f (t).2. If f is bounded, piecewise continuous and periodic with
period T , then
L{
f (t)}
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 16: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/16.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt =
∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt =
∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du =
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 17: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/17.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt =
∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt =
∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du =
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 18: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/18.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt =
∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du =
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 19: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/19.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt =
∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt =
∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du =
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 20: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/20.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt =
∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt
=∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du =
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 21: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/21.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt =
∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt =
∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du =
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 22: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/22.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt =
∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt =
∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du
=
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 23: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/23.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt =
∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt =
∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du =
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 24: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/24.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
How Did We Get That?
L{
f (t)}
=∫
∞
0e−stf (t) dt =
∞
∑n=0
∫ (n+1)T
nTe−stf (t) dt
=∞
∑n=0
∫ (n+1)T
nTe−s((t−nT)+nT
)f (t) dt =
∞
∑n=0
∫ T
0e−s(u+nT)f (u) du
=∞
∑n=0
e−nsT∫ T
0e−suf (u) du =
[∞
∑n=0
(e−sT)n
]∫ T
0e−stf (t) dt
=1
1− e−sT
∫ T
0e−stf (t) dt
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 25: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/25.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]=
11− e−sπ
[1
s2 +1+ e−πs 1
s2 +1
]=
11− e−sπ
[1+ e−πs] 1
s2 +1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 26: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/26.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]=
11− e−sπ
[1
s2 +1+ e−πs 1
s2 +1
]=
11− e−sπ
[1+ e−πs] 1
s2 +1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 27: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/27.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]=
11− e−sπ
[1
s2 +1+ e−πs 1
s2 +1
]=
11− e−sπ
[1+ e−πs] 1
s2 +1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 28: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/28.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]=
11− e−sπ
[1
s2 +1+ e−πs 1
s2 +1
]=
11− e−sπ
[1+ e−πs] 1
s2 +1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 29: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/29.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]=
11− e−sπ
[1
s2 +1+ e−πs 1
s2 +1
]=
11− e−sπ
[1+ e−πs] 1
s2 +1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 30: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/30.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]
=1
1− e−sπ
[1
s2 +1+ e−πs 1
s2 +1
]=
11− e−sπ
[1+ e−πs] 1
s2 +1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 31: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/31.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]=
11− e−sπ
[1
s2 +1
+ e−πs 1s2 +1
]=
11− e−sπ
[1+ e−πs] 1
s2 +1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 32: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/32.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]=
11− e−sπ
[1
s2 +1+ e−πs 1
s2 +1
]
=1
1− e−sπ
[1+ e−πs] 1
s2 +1
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 33: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/33.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
L{|sin(t)|
}=
11− e−sπ
∫π
0e−st∣∣sin(t)
∣∣ dt
=1
1− e−sπ
∫π
0e−st sin(t) dt
=1
1− e−sπ
∫∞
0e−st(1−U (t−π)
)sin(t) dt
=1
1− e−sπ
[L{
sin(t)}
+L{U (t−π)sin(t−π)
}]=
11− e−sπ
[1
s2 +1+ e−πs 1
s2 +1
]=
11− e−sπ
[1+ e−πs] 1
s2 +1Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 34: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/34.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
3sY +2Y =1
1− e−sπ
[1+ e−πs] 1
s2 +1
Y =1
1− e−sπ
[1+ e−πs] 1
(s2 +1)(3s+2)
=∞
∑n=0
(e−πs)n [1+ e−πs] 1
(s2 +1)(3s+2)
=
[∞
∑n=0
(e−πs)n +
∞
∑n=0
(e−πs)n+1
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 35: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/35.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
3sY +2Y
=1
1− e−sπ
[1+ e−πs] 1
s2 +1
Y =1
1− e−sπ
[1+ e−πs] 1
(s2 +1)(3s+2)
=∞
∑n=0
(e−πs)n [1+ e−πs] 1
(s2 +1)(3s+2)
=
[∞
∑n=0
(e−πs)n +
∞
∑n=0
(e−πs)n+1
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 36: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/36.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
3sY +2Y =1
1− e−sπ
[1+ e−πs] 1
s2 +1
Y =1
1− e−sπ
[1+ e−πs] 1
(s2 +1)(3s+2)
=∞
∑n=0
(e−πs)n [1+ e−πs] 1
(s2 +1)(3s+2)
=
[∞
∑n=0
(e−πs)n +
∞
∑n=0
(e−πs)n+1
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 37: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/37.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
3sY +2Y =1
1− e−sπ
[1+ e−πs] 1
s2 +1
Y =1
1− e−sπ
[1+ e−πs] 1
(s2 +1)(3s+2)
=∞
∑n=0
(e−πs)n [1+ e−πs] 1
(s2 +1)(3s+2)
=
[∞
∑n=0
(e−πs)n +
∞
∑n=0
(e−πs)n+1
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 38: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/38.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
3sY +2Y =1
1− e−sπ
[1+ e−πs] 1
s2 +1
Y =1
1− e−sπ
[1+ e−πs] 1
(s2 +1)(3s+2)
=∞
∑n=0
(e−πs)n [1+ e−πs] 1
(s2 +1)(3s+2)
=
[∞
∑n=0
(e−πs)n +
∞
∑n=0
(e−πs)n+1
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 39: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/39.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
3sY +2Y =1
1− e−sπ
[1+ e−πs] 1
s2 +1
Y =1
1− e−sπ
[1+ e−πs] 1
(s2 +1)(3s+2)
=∞
∑n=0
(e−πs)n [1+ e−πs] 1
(s2 +1)(3s+2)
=
[∞
∑n=0
(e−πs)n +
∞
∑n=0
(e−πs)n+1
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 40: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/40.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
3sY +2Y =1
1− e−sπ
[1+ e−πs] 1
s2 +1
Y =1
1− e−sπ
[1+ e−πs] 1
(s2 +1)(3s+2)
=∞
∑n=0
(e−πs)n [1+ e−πs] 1
(s2 +1)(3s+2)
=
[∞
∑n=0
(e−πs)n +
∞
∑n=0
(e−πs)n+1
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 41: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/41.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0
1(s2 +1)(3s+2)
=As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 42: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/42.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 43: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/43.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)
s =−23
: 1 = C(
49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 44: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/44.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3:
1 = C(
49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 45: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/45.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 46: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/46.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C
, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 47: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/47.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 48: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/48.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 :
1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 49: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/49.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1
= 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 50: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/50.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13
, B =213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 51: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/51.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 52: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/52.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 :
1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 53: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/53.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2
= 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 54: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/54.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 55: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/55.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 56: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/56.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 57: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/57.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 58: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/58.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 59: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/59.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)
=1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 60: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/60.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 01
(s2 +1)(3s+2)=
As+Bs2 +1
+C
3s+2
1 = (As+B)(3s+2)+C(s2 +1
)s =−2
3: 1 = C
(49
+1)
=139
C, C =913
s = 0 : 1 = B ·2+C ·1 = 2B+9
13, B =
213
s = 1 : 1 = (A+B) ·5+C ·2 = 5A+1013
+1813
, A =− 313
1(s2 +1)(3s+2)
=1
13
(−3s+2s2 +1
+9
3s+2
)=
113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 61: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/61.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
Y =
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[∞
∑n=0
e−nπs +∞
∑n=1
e−nπs
]113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[1+2
∞
∑n=1
e−nπs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 62: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/62.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
Y =
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[∞
∑n=0
e−nπs +∞
∑n=1
e−nπs
]113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[1+2
∞
∑n=1
e−nπs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 63: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/63.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
Y =
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[∞
∑n=0
e−nπs +∞
∑n=1
e−nπs
]113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[1+2
∞
∑n=1
e−nπs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 64: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/64.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
Y =
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[∞
∑n=0
e−nπs +∞
∑n=1
e−nπs
]113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[1+2
∞
∑n=1
e−nπs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 65: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/65.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem3y′+2y =
∣∣sin(t)∣∣, y(0) = 0
Y =
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
(s2 +1)(3s+2)
=
[∞
∑n=0
e−nπs +∞
∑n=0
e−(n+1)πs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[∞
∑n=0
e−nπs +∞
∑n=1
e−nπs
]113
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
=
[1+2
∞
∑n=1
e−nπs
]1
13
(−3
ss2 +1
+21
s2 +1+3
1s+ 2
3
)
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 66: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/66.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0
y =113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U(t−nπ
)[−3cos(t)+2sin(t)+3e−
23 t]
t→t−nπ
=113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−3cos(t−nπ)+2sin(t−nπ)+3e−
23 (t−nπ)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 67: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/67.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0
y =113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U(t−nπ
)[−3cos(t)+2sin(t)+3e−
23 t]
t→t−nπ
=113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−3cos(t−nπ)+2sin(t−nπ)+3e−
23 (t−nπ)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 68: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/68.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Solve the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0
y =113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U(t−nπ
)[−3cos(t)+2sin(t)+3e−
23 t]
t→t−nπ
=113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−3cos(t−nπ)+2sin(t−nπ)+3e−
23 (t−nπ)
]
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 69: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/69.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Does y =113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−3cos(t−nπ)+2sin(t−nπ)+3e−
23 (t−nπ)
]Really Solve
the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0?
y(0) =113
[−3cos(0)+2sin(0)+3e−
23 0]
+213
∞
∑n=1
U (0−nπ)[−3cos(0−nπ)+2sin(0−nπ)+3e−
23 (0−nπ)
]= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 70: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/70.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Does y =113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−3cos(t−nπ)+2sin(t−nπ)+3e−
23 (t−nπ)
]Really Solve
the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0?
y(0)
=113
[−3cos(0)+2sin(0)+3e−
23 0]
+213
∞
∑n=1
U (0−nπ)[−3cos(0−nπ)+2sin(0−nπ)+3e−
23 (0−nπ)
]= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 71: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/71.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Does y =113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−3cos(t−nπ)+2sin(t−nπ)+3e−
23 (t−nπ)
]Really Solve
the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0?
y(0) =113
[−3cos(0)+2sin(0)+3e−
23 0]
+213
∞
∑n=1
U (0−nπ)[−3cos(0−nπ)+2sin(0−nπ)+3e−
23 (0−nπ)
]
= 0√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 72: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/72.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Does y =113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−3cos(t−nπ)+2sin(t−nπ)+3e−
23 (t−nπ)
]Really Solve
the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0?
y(0) =113
[−3cos(0)+2sin(0)+3e−
23 0]
+213
∞
∑n=1
U (0−nπ)[−3cos(0−nπ)+2sin(0−nπ)+3e−
23 (0−nπ)
]= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 73: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/73.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Does y =113
[−3cos(t)+2sin(t)+3e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−3cos(t−nπ)+2sin(t−nπ)+3e−
23 (t−nπ)
]Really Solve
the Initial Value Problem 3y′+2y =∣∣sin(t)
∣∣, y(0) = 0?
y(0) =113
[−3cos(0)+2sin(0)+3e−
23 0]
+213
∞
∑n=1
U (0−nπ)[−3cos(0−nπ)+2sin(0−nπ)+3e−
23 (0−nπ)
]= 0
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 74: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/74.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
y′ =1
13
[3sin(t)+2cos(t)−2e−
23 t]
+213
∞
∑n=1
U (t−nπ)[3sin(t−nπ)+2cos(t−nπ)−2e−
23 (t−nπ)
]3y′+2y =
113
[9sin(t)+6cos(t)−6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[9sin(t−nπ)+6cos(t−nπ)−6e−
23 (t−nπ)
]+
113
[−6cos(t)+4sin(t)+6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−6cos(t−nπ)+4sin(t−nπ)+6e−
23 (t−nπ)
]=
113
[9sin(t)+4sin(t)]+2
13
∞
∑n=1
U (t−nπ) [9sin(t−nπ)+4sin(t−nπ)]
= sin(t)+2∞
∑n=1
U (t−nπ)sin(t−nπ) =∣∣sin(t)
∣∣ √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 75: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/75.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
y′ =1
13
[3sin(t)+2cos(t)−2e−
23 t]
+213
∞
∑n=1
U (t−nπ)[3sin(t−nπ)+2cos(t−nπ)−2e−
23 (t−nπ)
]
3y′+2y =1
13
[9sin(t)+6cos(t)−6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[9sin(t−nπ)+6cos(t−nπ)−6e−
23 (t−nπ)
]+
113
[−6cos(t)+4sin(t)+6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−6cos(t−nπ)+4sin(t−nπ)+6e−
23 (t−nπ)
]=
113
[9sin(t)+4sin(t)]+2
13
∞
∑n=1
U (t−nπ) [9sin(t−nπ)+4sin(t−nπ)]
= sin(t)+2∞
∑n=1
U (t−nπ)sin(t−nπ) =∣∣sin(t)
∣∣ √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 76: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/76.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
y′ =1
13
[3sin(t)+2cos(t)−2e−
23 t]
+213
∞
∑n=1
U (t−nπ)[3sin(t−nπ)+2cos(t−nπ)−2e−
23 (t−nπ)
]3y′+2y =
113
[9sin(t)+6cos(t)−6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[9sin(t−nπ)+6cos(t−nπ)−6e−
23 (t−nπ)
]
+113
[−6cos(t)+4sin(t)+6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−6cos(t−nπ)+4sin(t−nπ)+6e−
23 (t−nπ)
]=
113
[9sin(t)+4sin(t)]+2
13
∞
∑n=1
U (t−nπ) [9sin(t−nπ)+4sin(t−nπ)]
= sin(t)+2∞
∑n=1
U (t−nπ)sin(t−nπ) =∣∣sin(t)
∣∣ √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 77: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/77.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
y′ =1
13
[3sin(t)+2cos(t)−2e−
23 t]
+213
∞
∑n=1
U (t−nπ)[3sin(t−nπ)+2cos(t−nπ)−2e−
23 (t−nπ)
]3y′+2y =
113
[9sin(t)+6cos(t)−6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[9sin(t−nπ)+6cos(t−nπ)−6e−
23 (t−nπ)
]+
113
[−6cos(t)+4sin(t)+6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−6cos(t−nπ)+4sin(t−nπ)+6e−
23 (t−nπ)
]
=1
13[9sin(t)+4sin(t)]+
213
∞
∑n=1
U (t−nπ) [9sin(t−nπ)+4sin(t−nπ)]
= sin(t)+2∞
∑n=1
U (t−nπ)sin(t−nπ) =∣∣sin(t)
∣∣ √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 78: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/78.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
y′ =1
13
[3sin(t)+2cos(t)−2e−
23 t]
+213
∞
∑n=1
U (t−nπ)[3sin(t−nπ)+2cos(t−nπ)−2e−
23 (t−nπ)
]3y′+2y =
113
[9sin(t)+6cos(t)−6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[9sin(t−nπ)+6cos(t−nπ)−6e−
23 (t−nπ)
]+
113
[−6cos(t)+4sin(t)+6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−6cos(t−nπ)+4sin(t−nπ)+6e−
23 (t−nπ)
]=
113
[9sin(t)+4sin(t)]+2
13
∞
∑n=1
U (t−nπ) [9sin(t−nπ)+4sin(t−nπ)]
= sin(t)+2∞
∑n=1
U (t−nπ)sin(t−nπ) =∣∣sin(t)
∣∣ √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 79: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/79.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
y′ =1
13
[3sin(t)+2cos(t)−2e−
23 t]
+213
∞
∑n=1
U (t−nπ)[3sin(t−nπ)+2cos(t−nπ)−2e−
23 (t−nπ)
]3y′+2y =
113
[9sin(t)+6cos(t)−6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[9sin(t−nπ)+6cos(t−nπ)−6e−
23 (t−nπ)
]+
113
[−6cos(t)+4sin(t)+6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−6cos(t−nπ)+4sin(t−nπ)+6e−
23 (t−nπ)
]=
113
[9sin(t)+4sin(t)]+2
13
∞
∑n=1
U (t−nπ) [9sin(t−nπ)+4sin(t−nπ)]
= sin(t)+2∞
∑n=1
U (t−nπ)sin(t−nπ)
=∣∣sin(t)
∣∣ √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 80: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/80.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
y′ =1
13
[3sin(t)+2cos(t)−2e−
23 t]
+213
∞
∑n=1
U (t−nπ)[3sin(t−nπ)+2cos(t−nπ)−2e−
23 (t−nπ)
]3y′+2y =
113
[9sin(t)+6cos(t)−6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[9sin(t−nπ)+6cos(t−nπ)−6e−
23 (t−nπ)
]+
113
[−6cos(t)+4sin(t)+6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−6cos(t−nπ)+4sin(t−nπ)+6e−
23 (t−nπ)
]=
113
[9sin(t)+4sin(t)]+2
13
∞
∑n=1
U (t−nπ) [9sin(t−nπ)+4sin(t−nπ)]
= sin(t)+2∞
∑n=1
U (t−nπ)sin(t−nπ) =∣∣sin(t)
∣∣
√
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 81: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/81.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
y′ =1
13
[3sin(t)+2cos(t)−2e−
23 t]
+213
∞
∑n=1
U (t−nπ)[3sin(t−nπ)+2cos(t−nπ)−2e−
23 (t−nπ)
]3y′+2y =
113
[9sin(t)+6cos(t)−6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[9sin(t−nπ)+6cos(t−nπ)−6e−
23 (t−nπ)
]+
113
[−6cos(t)+4sin(t)+6e−
23 t]
+213
∞
∑n=1
U (t−nπ)[−6cos(t−nπ)+4sin(t−nπ)+6e−
23 (t−nπ)
]=
113
[9sin(t)+4sin(t)]+2
13
∞
∑n=1
U (t−nπ) [9sin(t−nπ)+4sin(t−nπ)]
= sin(t)+2∞
∑n=1
U (t−nπ)sin(t−nπ) =∣∣sin(t)
∣∣ √
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 82: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/82.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Comparing Output to Input
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 83: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/83.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Comparing Output to Input
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions
![Page 84: Laplace Transforms of Periodic Functions · 2008. 11. 6. · Periodic Functions 1. A function f is periodic with period T >0 if and only if for all t we have f(t+T)=f(t). 2. If f](https://reader035.vdocument.in/reader035/viewer/2022071105/5fdeb6468a36da1042306238/html5/thumbnails/84.jpg)
logo1
Transforms and New Formulas An Example Double Check Visualization
Comparing Output to Input
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Laplace Transforms of Periodic Functions