Non-Homogeneous Equations

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)