Non-Homogeneous Equations Method of Undetermined Coefficients

We Know How To Solve Homogeneous 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? as a guide For the Non-homogeneous equation, guess a different form of solution.

Example

Example Use to guess form of a solution suggests that (This is the undetermined coefficient)

to guess form of a solution Example Use to guess form of a solution suggests that Then:

Example suggests that Then: Plugging In:

Example suggests that Then: Plugging In:

Example suggests that Then: Plugging In:

Example suggests that Then: Plugging In: These are the same

Example suggests that Then: Plugging In: Specific Solution:

Method of Undetermined Coefficients Use as a guide Guess that specific solution takes the form: (This is the undetermined coefficient)

Method of Undetermined Coefficients Use as a guide Guess that specific solution takes the form: Plug in to differential equation Solve for

Method of Undetermined Coefficients Guess that specific solution takes the form: Plug in to differential equation Solve for Determining the right Depends 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!

But Adding or Multiplying By a Constant General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in:

But Adding or Multiplying By a Constant General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in:

But Adding or Multiplying By a Constant General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in:

But Adding or Multiplying By a Constant General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in:

But Adding or Multiplying By a Constant General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in: These are the same!

General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you add a constant And substitute in: No help for finding General Solutions!

General Solutions But Adding or Multiplying By a Constant Breaks the Solution! If you multiply by a constant And substitute in (exercise - try it): No help for finding General Solutions!

General Solutions So how do we find general solutions? Go back to the homogeneous case Find general solution, i.e. where (The “h” is for “homogeneous”)

General Solutions For If is a specific solution to the non-homogeneous equation And is the general solution to the homogeneous equation Then Is a general solution to the homogeneous equation

(General Homogeneous Solution) General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

(General Homogeneous Solution) General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

(General Homogeneous Solution) General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

(General Homogeneous Solution) General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

(General Homogeneous Solution) General Solutions (Specific Solution) (General Homogeneous Solution) Plug in

General Solutions (Specific Solution) (General Homogeneous Solution) Plug in So it is a (General) Solution

Example Specific 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 Coefficient Guesses (“Ansatz”) Form or or Times anything above Times Corresponding Form

Divide and Conquer If Can Find Specific Solutions And Their Sum Will Be A Specific Solution To (The Logic Is Identical To Why Is A General Solution)