Variation of
Parameters
Method for Non-Homogeneous Equations
Recall Non-Homogeneous
Linear Equation
Solve by finding specific solution
And finding general solution to the
Homogeneous Equation
The General Solution takes the
form:
Last Time
We found specific solutions to the
Non-Homogeneous Equation
Using Undetermined Coefficients:
Guess that specific solution takes the form:
Plug in to differential equation
Solve
for
Undetermined Coefficients is
Only Appropriate for Certain g(t)
or
or
Times anything above
Undetermined Coefficients is
Only Appropriate for Certain g(t)
What if
doesn’t take that form?
Use a more general technique
Variation of
Parameters
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
Variation of Parameters
Search for a specific solution of the form
Again, plug in to the equation,
find conditions on
So
:
So… many…
terms…
An Important Aside
Linear Algebra Question:
How many
solutions?
(One equation, Two unknowns…)
Infinite!
An Important Aside
Linear Algebra Question:
But if we add one condition
How many
solutions?
Just one.
An Important Aside
This holds for
Must satisfy
(One equation, two unknowns)
Infinite different
Will satisfy the equation
An Important Aside
This holds for
Infinite different
Will satisfy the equation
Let’s add a condition
Now,
Are uniquely defined
Variation of Parameters
Search for a specific solution of the form
Again, plug in to the equation,
find conditions on
So
:
So… many…
terms…
Variation of Parameters
Search for a specific solution of the form
Add the condition
So
:
So… many…
terms…
Variation of Parameters
Search for a specific solution of the form
Add the condition
So
:
So… many…
terms…
Variation of Parameters
Search for a specific solution of the form
Add the condition
So
:
So… many…
terms…
Variation of Parameters
Search for a specific solution of the form
Add the condition
So
:
So… many…
terms…
Variation of Parameters
Search for a specific solution of the form
Add the condition
So
:
So… many…
terms…
Variation of Parameters
Search for a specific solution of the form
Add the condition
So
:
Not so bad…
Variation of Parameters
Search for a specific solution of the form
Add the condition
So
:
Variation of Parameters
Search for a specific solution of the form
Add the condition
So
:
Variation of Parameters
Plug Into Equation
Variation of Parameters
Distribute Coefficients
Variation of Parameters
Factor out
and
Variation of Parameters
Factor out
and
But remember,
Solve the homogeneous equation
Which means….
Variation of Parameters
Simplify
But remember,
Solve the homogeneous equation
Which means….
Variation of Parameters
Simplify
But remember,
Solve the homogeneous equation
Which means….
Variation of Parameters
Simplify
But remember,
Solve the homogeneous equation
Which means….
Variation of Parameters
Simplify
But remember,
Solve the homogeneous equation
Which means….
Variation of Parameters
Simplify
But remember,
Solve the homogeneous equation
Which means….
Variation of Parameters
Simplify
But remember,
Solve the homogeneous equation
Which means….
Variation of Parameters
Simplify
Bringing Back Condition
Variation of Parameters
Simplify
Two
Equations
Two
Unknowns
Can Solve
and
For
Variation of Parameters
Simplify
Solutions
:
and
Wher
e
is the Wronskian of
and
Variation of Parameters
Simplify
Solutions
:
and
Variation of Parameters
General Solution Takes the
Form
Wher
and
e
Solve Homogeneous Equation
and
So to solve…
Find the general solution to the Homogeneous Equation
Solv
e
and
General
Solution:
Example
Find the general solution to the Homogeneous Equation
General Solution Takes the
Form
Wher
and
e
Solve Homogeneous Equation
and
Variation of Parameters
General Solution Takes the
Form
Wher
and
e
Solve Homogeneous Equation
and
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)
Questions?