1. Non-Homogeneous Equations
Method of Undetermined Coefficients

2. We Know How To Solve Homogeneous Equations
(With Constant Coefficients)
Find Roots of Characteristic Polynomial
Determine Appropriate General Solution

3. But what about Non-Homogeneous Equations?
Recall that we assumed the solution
For the homogeneous equation

4. But what about Non-Homogeneous Equations?
as a guide
For the Non-homogeneous equation,
guess a different form of solution.

5. Example

6. Example Use to guess form of a solution suggests that
(This is the undetermined coefficient)

7. to guess form of a solution
Example
Use
to guess form of a solution
suggests that
Then:

8. Example
suggests that
Then:
Plugging In:

9. Example
suggests that
Then:
Plugging In:

10. Example
suggests that
Then:
Plugging In:

11. Example
suggests that
Then:
Plugging In:
These are the same

12. Example
suggests that
Then:
Plugging In:
Specific Solution:

13. Method of Undetermined Coefficients
Use
as a guide
Guess that specific solution takes the form:
(This is the undetermined coefficient)

14. Method of Undetermined Coefficients
Use
as a guide
Guess that specific solution takes the form:
Plug in to differential equation
Solve for

15. Method of Undetermined Coefficients
Guess that specific solution takes the form:
Plug in to differential equation
Solve for
Determining the right
Depends on
(Will go through important cases later)

16. General Solutions
Undetermined Coefficients Gives one Specific Solution
But Adding or Multiplying By a Constant
Breaks the Solution!

17. But Adding or Multiplying By a Constant
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:

18. But Adding or Multiplying By a Constant
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:

19. But Adding or Multiplying By a Constant
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:

20. But Adding or Multiplying By a Constant
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:

21. But Adding or Multiplying By a Constant
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:
These are the same!

22. General Solutions But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:
No help for finding General Solutions!

23. General Solutions But Adding or Multiplying By a Constant
Breaks the Solution!
If you multiply by a constant
And substitute in (exercise - try it):
No help for finding General Solutions!

24. General Solutions So how do we find general solutions?
Go back to the homogeneous case
Find general solution, i.e.
where
(The “h” is for “homogeneous”)

25. General Solutions For If is a specific solution to the
non-homogeneous equation
And
is the general solution to the
homogeneous equation
Then
Is a general solution to the homogeneous equation

26. (General Homogeneous Solution)
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in

27. (General Homogeneous Solution)
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in

28. (General Homogeneous Solution)
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in

29. (General Homogeneous Solution)
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in

30. (General Homogeneous Solution)
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in

31. General Solutions (Specific Solution) (General Homogeneous Solution)
Plug in
So it is a (General) Solution

32. Example Specific Solution: Homogeneous Equation Has General Solution
(I assume you can determine this)
So the Non-Homogeneous Equation
Has General Solution

33. So to solve…

34. So to solve… Use Undetermined Coefficients to find a specific solution
Find the general solution
To the Homogeneous Equation

35. 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:

36. 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..

37. Questions?

38. Undetermined Coefficient Guesses (“Ansatz”)
Form or or
Times anything above
Times Corresponding Form

39. Divide and Conquer If Can Find Specific Solutions And Their Sum
Will Be A Specific Solution To
(The Logic Is Identical To Why Is A General Solution)