Variation of Parameters Method for Non-Homogeneous Equations.

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?