second order equations. so far… we have been solving linear and nonlinear first order equations....
TRANSCRIPT
![Page 1: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/1.jpg)
Second Order Equations
![Page 2: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/2.jpg)
So Far…We have been solving linear
and nonlinear first order
equations.Those days are
over.
Today, we will start examining
second order linear equations.
(Sort of)
![Page 3: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/3.jpg)
Bad (or Good?) NewsHigher order equations are
MUCH more difficult to solve than
first order equations.
So we start simple, and work our way
up.
Linear equations are hard.Nonlinear equations
are crazy hard.
![Page 4: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/4.jpg)
Linear Second Order Equations
The form of a second order linear equation:
Not simple enough!
Still not simple enough!
Homogenous
equation.
![Page 5: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/5.jpg)
Linear Second Order Equations
The form of a second order linear equation:
Not simple enough!
Still not simple enough!
![Page 6: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/6.jpg)
Linear Second Order Equations
The form of a second order linear equation:
Not simple enough!
Still not simple enough!
Okay, that’s a good place to start.
![Page 7: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/7.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
Side note:
We already know one specific solution to this
equation.
Equilibrium Solution!
Always check for equilibrium solutions!!!!
![Page 8: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/8.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
We will consider solutions of the form:
Why?
There is a reason.
We will discuss it later in the course.
![Page 9: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/9.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
We will consider solutions of the form:
![Page 10: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/10.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
We will consider solutions of the form:
Then:
![Page 11: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/11.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
We will consider solutions of the form:
Then:
![Page 12: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/12.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
We will consider solutions of the form:
Then:
![Page 13: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/13.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
We will consider solutions of the form:
Then:
![Page 14: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/14.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
Factoring out:
One of these two must be 0
![Page 15: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/15.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
Factoring out:
Can’t be 0
![Page 16: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/16.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
Factoring out:
Must be 0
![Page 17: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/17.jpg)
Second Order Linear Homogenous Equations with
Constant Coefficients
Factoring out:
So
This is called the characteristic equation.
![Page 18: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/18.jpg)
So to solve…
Find the characteristic equation
Find(Either factor or use the quadratic
equation)Write the general
solution:
![Page 19: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/19.jpg)
The catch…Because there’s always a
catch….
FindTwo
solutions!
Three possibilities
or
![Page 20: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/20.jpg)
So to solve…Find the characteristic
equation
FindWrite the general solution:
and
![Page 21: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/21.jpg)
Specific SolutionsFor first order equations, need 1 initial
condition
For second order equations, need 2 initial conditions
For order N equations, need N initial conditions
![Page 22: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/22.jpg)
Specific SolutionFor second order equations, need 2 initial
conditions
To find specific solution:
Plug in Initial Conditions:
![Page 23: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/23.jpg)
Specific SolutionFor second order equations, need 2 initial
conditions
To find specific solution:
Plug in Initial Conditions:
Solve for:
and
![Page 24: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/24.jpg)
So to solve…Find the characteristic
equationFindWrite the general solution:
and
Plug in Initial Conditions:
Solve for:
and
![Page 25: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/25.jpg)
ExampleFind the characteristic
equationFind
and
and
![Page 26: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/26.jpg)
ExampleFind the characteristic
equationFind
andan
dWrite the general solution:
Plug in Initial Conditions:
Solve for
andan
d
![Page 27: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/27.jpg)
ExampleFind the characteristic
equationFind
andan
dWrite the general solution:
Plug in Initial Conditions:Solve
forandan
d
![Page 28: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/28.jpg)
Summary• Higher Order Equations are Difficult To Solve
• Can solve Second Order Homogeneous Linear Equations with Constant Coefficients Using Characteristic Equations
• Need 2 Initial Conditions for Second Order Equations
• In general, need N initial conditions for Nth order equation
![Page 29: Second Order Equations. So Far… We have been solving linear and nonlinear first order equations. Those days are over. Today, we will start examining second](https://reader035.vdocument.in/reader035/viewer/2022062717/56649e215503460f94b0dffd/html5/thumbnails/29.jpg)
Questions?