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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 TimeWe found specific solutions to
theNon-Homogeneous Equation
Using Undetermined Coefficients:
Guess that specific solution takes the form:
Plug in to differential equationSolve
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 AsideLinear Algebra
Question:
How many solutions?(One equation, Two
unknowns…)Infinite!
An Important AsideLinear Algebra
Question:
How many solutions?Just
one.
But if we add one condition
An Important AsideThis holds
for
Must satisfy
(One equation, two unknowns)
Infinite different
Will satisfy the equation
An Important AsideThis 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
But remember,
Solve the homogeneous equation
Which means….
Simplify
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
Variation of Parameters
But remember,
Solve the homogeneous equation
Which means….
Simplify
Variation of Parameters
Simplify
Bringing Back Condition
Variation of Parameters
Simplify
Two EquationsTwo UnknownsCan Solve
Forand
Variation of Parameters
Simplify
Solutions:and
Where
is the Wronskian of
and
Variation of Parameters
Simplify
Solutions:
and
Variation of Parameters
General Solution Takes the Form
Where
andSolve Homogeneous
Equation and
So to solve…
Find the general solution to the Homogeneous Equation
Solve
and
General Solution:
ExampleFind the general solution to the Homogeneous
EquationGeneral Solution Takes the Form
Where
andSolve Homogeneous
Equation and
Variation of Parameters
General Solution Takes the Form
Where
andSolve 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?