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 Times anything above

Undetermined Coefficients is Only Appropriate for Certain g(t) What ifdoesn’t take that form? Use a more general technique 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: How many solutions? Just one. But if we add one condition

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 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 Equations Two Unknowns Can Solve For and

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 e and 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 e and Solve Homogeneous Equation and

Variation of Parameters General Solution Takes the Form Wher e and 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?