Non-Homogeneous
Equations
Method of Undetermined Coefficients
We Know How To Solve
Homogeneous Equations
(With Constant Coefficients)
Find Roots of Characteristic Polynomial
Determine Appropriate General Solution
But what about
Non-Homogeneous Equations?
Recall that we assumed the solution
For the homogeneous equation
But what about
Non-Homogeneous Equations?
Use
as a guide
For the Non-homogeneous equation,
guess a different form of solution.
Example
Example
Use
to guess form of a solution
suggests that
(This is the undetermined coefficient)
Example
Use
to guess form of a solution
suggests that
Then
:
Example
suggests that
Then
:
Plugging In:
Example
suggests that
Then
:
Plugging In:
Example
suggests that
Then
:
Plugging In:
Example
suggests that
Then
:
Plugging In:
These are the
same
Example
suggests that
Then
:
Plugging In:
Specific Solution:
Method of
Undetermined Coefficients
Use
as a guide
Guess that specific solution takes the form:
(This is the undetermined coefficient)
Method of
Undetermined Coefficients
Use
as a guide
Guess that specific solution takes the form:
Plug in to differential equation
Solve
for
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)
General Solutions
Undetermined Coefficients Gives one Specific Solution
But Adding or Multiplying By a Constant
Breaks the Solution!
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:
General Solutions
But Adding or Multiplying By a Constant
Breaks the Solution!
If you add a constant
And substitute in:
These are the same!
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!
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!
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”)
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
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in
General Solutions
(Specific Solution)
(General Homogeneous Solution)
Plug in
So it is a (General)
Solution
Example
Specific Solution:
Homogeneous Equation
Has General Solution
(I assume you can determine this)
So the Non-Homogeneous Equation
Has General Solution
So to solve…
So to solve…
Use Undetermined Coefficients
to find a specific solution
Find the general solution
To the Homogeneous Equation
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:
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..
Questions?
Undetermined Coefficient
Guesses (“Ansatz”)
For
m
or
or
Times anything above
Times Corresponding Form
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)