Non-Homogeneous Equations Method of Undetermined Coefficients
Jan 01, 2016
Non-Homogeneous Equations
Method of Undetermined Coefficients
We Know How To SolveHomogeneous 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?
For the Non-homogeneous equation,
guess a different form of solution.
Use
as a guide
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 thatThen
:
Example
suggests thatThen
:
Plugging In:
Example
suggests thatThen
:
Plugging In:
Example
suggests thatThen
:
Plugging In:
Example
suggests thatThen
:
Plugging In:
These are the same
Example
suggests thatThen
:
Plugging In:
Specific Solution:
Method of Undetermined
Coefficients
Guess that specific solution takes the form:
Use
as a guide
(This is the undetermined coefficient)
Method of Undetermined
Coefficients
Guess that specific solution takes the form:
Use
as a guide
Plug in to differential equationSolve
for
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)
General Solutions
Undetermined Coefficients Gives one Specific Solution
But Adding or Multiplying By a Constant
Breaks the Solution!
General SolutionsBut Adding or Multiplying By a
Constant Breaks the Solution!
If you add a constant
And substitute in:
General Solutions
If you add a constant
And substitute in:
But Adding or Multiplying By a Constant
Breaks the Solution!
General Solutions
If you add a constant
And substitute in:
But Adding or Multiplying By a Constant
Breaks the Solution!
General Solutions
If you add a constant
And substitute in:
But Adding or Multiplying By a Constant
Breaks the Solution!
General Solutions
If you add a constant
And substitute in:
These are the same!
But Adding or Multiplying By a Constant
Breaks the Solution!
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!
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!
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
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
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
ExampleSpecific 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 CoefficientGuesses (“Ansatz”)
Form
or
or
Times anything above
Times Corresponding Form
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)