Variation of Parameters Method for Non-Homogeneous Equations.
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Transcript of Variation of Parameters Method for Non-Homogeneous Equations.
![Page 1: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/1.jpg)
Variation of Parameters
Method for Non-Homogeneous Equations
![Page 2: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/2.jpg)
Recall Non-Homogeneous Linear Equation
Solve by finding specific solution
And finding general solution to the
Homogeneous Equation
The General Solution takes the form:
![Page 3: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/3.jpg)
Last TimeWe found specific solutions to
theNon-Homogeneous Equation
Using Undetermined Coefficients:
Guess that specific solution takes the form:
Plug in to differential equationSolve
for
![Page 4: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/4.jpg)
Undetermined Coefficients is Only Appropriate for Certain
g(t)
or
or
Times anything above
![Page 5: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/5.jpg)
Undetermined Coefficients is Only Appropriate for Certain
g(t)
What if
doesn’t take that form?
Use a more general technique
Variation of Parameters
![Page 6: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/6.jpg)
Variation of Parameters
Suppose we know the general solution
to the Homogeneous Equation
Which takes the form
Search for a specific solution of the form
![Page 7: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/7.jpg)
Variation of Parameters
Search for a specific solution of the form
Again, plug in to the equation,
find conditions on
So:
So… many… terms…
![Page 8: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/8.jpg)
An Important AsideLinear Algebra
Question:
How many solutions?(One equation, Two
unknowns…)Infinite!
![Page 9: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/9.jpg)
An Important AsideLinear Algebra
Question:
How many solutions?Just
one.
But if we add one condition
![Page 10: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/10.jpg)
An Important AsideThis holds
for
Must satisfy
(One equation, two unknowns)
Infinite different
Will satisfy the equation
![Page 11: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/11.jpg)
An Important AsideThis holds
for
Infinite different
Will satisfy the equation
Let’s add a condition
Now,Are uniquely
defined
![Page 12: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/12.jpg)
Variation of Parameters
Search for a specific solution of the form
Again, plug in to the equation,
find conditions on
So:
So… many… terms…
![Page 13: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/13.jpg)
Variation of Parameters
Search for a specific solution of the form
Add the condition
So:
So… many… terms…
![Page 14: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/14.jpg)
Variation of Parameters
Search for a specific solution of the form
Add the condition
So:
So… many… terms…
![Page 15: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/15.jpg)
Variation of Parameters
Search for a specific solution of the form
Add the condition
So:
So… many… terms…
![Page 16: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/16.jpg)
Variation of Parameters
Search for a specific solution of the form
Add the condition
So:
So… many… terms…
![Page 17: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/17.jpg)
Variation of Parameters
Search for a specific solution of the form
Add the condition
So:
So… many… terms…
![Page 18: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/18.jpg)
Variation of Parameters
Search for a specific solution of the form
Add the condition
So:
Not so bad…
![Page 19: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/19.jpg)
Variation of Parameters
Search for a specific solution of the form
Add the condition
So:
![Page 20: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/20.jpg)
Variation of Parameters
Search for a specific solution of the form
Add the condition
So:
![Page 21: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/21.jpg)
Variation of Parameters
Plug Into Equation
![Page 22: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/22.jpg)
Variation of Parameters
Distribute Coefficients
![Page 23: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/23.jpg)
Variation of Parameters
Factor out
and
![Page 24: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/24.jpg)
Variation of Parameters
Factor out
and
But remember,
Solve the homogeneous equation
Which means….
![Page 25: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/25.jpg)
Variation of Parameters
Simplify
But remember,
Solve the homogeneous equation
Which means….
![Page 26: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/26.jpg)
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
![Page 27: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/27.jpg)
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
![Page 28: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/28.jpg)
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
![Page 29: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/29.jpg)
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
![Page 30: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/30.jpg)
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
![Page 31: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/31.jpg)
Variation of Parameters
Simplify
Bringing Back Condition
![Page 32: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/32.jpg)
Variation of Parameters
Simplify
Two EquationsTwo UnknownsCan Solve
Forand
![Page 33: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/33.jpg)
Variation of Parameters
Simplify
Solutions:and
Where
is the Wronskian of
and
![Page 34: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/34.jpg)
Variation of Parameters
Simplify
Solutions:
and
![Page 35: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/35.jpg)
Variation of Parameters
General Solution Takes the Form
Where
andSolve Homogeneous
Equation and
![Page 36: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/36.jpg)
So to solve…
Find the general solution to the Homogeneous Equation
Solve
and
General Solution:
![Page 37: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/37.jpg)
ExampleFind the general solution to the Homogeneous
EquationGeneral Solution Takes the Form
Where
andSolve Homogeneous
Equation and
![Page 38: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/38.jpg)
Variation of Parameters
General Solution Takes the Form
Where
andSolve Homogeneous
Equation and
![Page 39: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/39.jpg)
Summary• Variation of Parameters Gives another
approach to solve Non-homogeneous linear ODEs
• From general solution of the Homogenous equation, find solution to Non-homogeneous equation
• It may be hard to integrate the parameter or find general solution of Homogeneous equation (which is why we don’t do this always)
![Page 40: Variation of Parameters Method for Non-Homogeneous Equations.](https://reader036.fdocuments.in/reader036/viewer/2022062301/56649e885503460f94b8d32e/html5/thumbnails/40.jpg)
Questions?