Non-Homogeneous Equations
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![Page 1: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/1.jpg)
Non-Homogeneous Equations
Method of Undetermined Coefficients
![Page 2: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/2.jpg)
We Know How To SolveHomogeneous Equations
(With Constant Coefficients)
Find Roots of Characteristic Polynomial
Determine Appropriate General Solution
![Page 3: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/3.jpg)
But what about Non-Homogeneous
Equations?
Recall that we assumed the solution
For the homogeneous equation
![Page 4: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/4.jpg)
But what about Non-Homogeneous
Equations?
For the Non-homogeneous equation,
guess a different form of solution.
Use
as a guide
![Page 5: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/5.jpg)
Example
![Page 6: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/6.jpg)
Example
Use
to guess form of a solution
suggests that
(This is the undetermined coefficient)
![Page 7: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/7.jpg)
Example
Use
to guess form of a solution
suggests thatThen
:
![Page 8: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/8.jpg)
Example
suggests thatThen
:
Plugging In:
![Page 9: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/9.jpg)
Example
suggests thatThen
:
Plugging In:
![Page 10: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/10.jpg)
Example
suggests thatThen
:
Plugging In:
![Page 11: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/11.jpg)
Example
suggests thatThen
:
Plugging In:
These are the same
![Page 12: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/12.jpg)
Example
suggests thatThen
:
Plugging In:
Specific Solution:
![Page 13: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/13.jpg)
Method of Undetermined
Coefficients
Guess that specific solution takes the form:
Use
as a guide
(This is the undetermined coefficient)
![Page 14: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/14.jpg)
Method of Undetermined
Coefficients
Guess that specific solution takes the form:
Use
as a guide
Plug in to differential equationSolve
for
![Page 15: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/15.jpg)
Method of Undetermined
Coefficients
Guess that specific solution takes the form:
Plug in to differential equationSolve
forDetermining the
rightDepends on
(Will go through important cases later)
![Page 16: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/16.jpg)
General Solutions
Undetermined Coefficients Gives one Specific Solution
But Adding or Multiplying By a Constant
Breaks the Solution!
![Page 17: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/17.jpg)
General SolutionsBut Adding or Multiplying By a
Constant Breaks the Solution!
If you add a constant
And substitute in:
![Page 18: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/18.jpg)
General Solutions
If you add a constant
And substitute in:
But Adding or Multiplying By a Constant
Breaks the Solution!
![Page 19: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/19.jpg)
General Solutions
If you add a constant
And substitute in:
But Adding or Multiplying By a Constant
Breaks the Solution!
![Page 20: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/20.jpg)
General Solutions
If you add a constant
And substitute in:
But Adding or Multiplying By a Constant
Breaks the Solution!
![Page 21: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/21.jpg)
General Solutions
If you add a constant
And substitute in:
These are the same!
But Adding or Multiplying By a Constant
Breaks the Solution!
![Page 22: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/22.jpg)
General Solutions
If you add a constant
And substitute in:
No help for finding General Solutions!
But Adding or Multiplying By a Constant
Breaks the Solution!
![Page 23: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/23.jpg)
General Solutions
If you multiply by a constant
And substitute in (exercise - try it):
No help for finding General Solutions!
But Adding or Multiplying By a Constant
Breaks the Solution!
![Page 24: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/24.jpg)
General Solutions
So how do we find general solutions?
Go back to the homogeneous case
Find general solution, i.e.
(The “h” is for “homogeneous”)
where
![Page 25: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/25.jpg)
General SolutionsFor
Ifis a specific solution to the
non-homogeneous equationAn
dis the general solution to the
homogeneous equationThen
Is a general solution to the homogeneous equation
![Page 26: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/26.jpg)
General Solutions(Specific Solution)(General Homogeneous
Solution)
Plug in
![Page 27: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/27.jpg)
General Solutions(Specific Solution)(General Homogeneous
Solution)
Plug in
![Page 28: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/28.jpg)
General Solutions(Specific Solution)(General Homogeneous
Solution)
Plug in
![Page 29: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/29.jpg)
General Solutions(Specific Solution)(General Homogeneous
Solution)
Plug in
![Page 30: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/30.jpg)
General Solutions(Specific Solution)(General Homogeneous
Solution)
Plug in
![Page 31: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/31.jpg)
General Solutions(Specific Solution)(General Homogeneous
Solution)
Plug in
So it is a (General) Solution
![Page 32: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/32.jpg)
ExampleSpecific Solution:Homogeneous
Equation
Has General Solution(I assume you can determine
this)
So the Non-Homogeneous Equation
Has General Solution
![Page 33: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/33.jpg)
So to solve…
![Page 34: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/34.jpg)
So to solve…
Use Undetermined Coefficients
to find a specific solution
Find the general solution
To the Homogeneous Equation
![Page 35: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/35.jpg)
So to solve…
Use Undetermined Coefficients
to find a specific solution
Find the general solution
To the Homogeneous Equation
The General Solution takes the form:
![Page 36: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/36.jpg)
Summary
• Method of Undetermined Coefficients Gives a Specific Solution For Non-Homogenous Equations
• General Solution comes from General Solution of Homogeneous Equation
• We will discuss Undetermined Coefficients More Next..
![Page 37: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/37.jpg)
Questions?
![Page 38: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/38.jpg)
Undetermined CoefficientGuesses (“Ansatz”)
Form
or
or
Times anything above
Times Corresponding Form
![Page 39: Non-Homogeneous Equations](https://reader034.fdocuments.net/reader034/viewer/2022050617/56813198550346895d980cc9/html5/thumbnails/39.jpg)
Divide and ConquerIf
Can Find Specific Solutions
And Their Sum
Will Be A Specific Solution To
(The Logic Is Identical To Why Is A General Solution)