Math 240: Linear Differential Equations - Undetermined Coefficients, Lecture notes of Differential Equations

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Math 240: Linear Differential Equations

Ryan Blair

University of Pennsylvania

Tuesday February 15, 2011

Outline

(^1) Review

(^2) Today’s Goals

(^3) Undetermined Coefficients

Review

Auxiliary Equations

Given a linear homogeneous constant-coefficient differential equation

an d

n (^) y dxn^ +^ an−^1

dn−^1 y dxn−^1 +^ ...a^1

dy dx +^ a^0 y^ = 0,

the Auxiliary Equation is

anmn^ + an− 1 mn−^1 + ...a 1 m + a 0 = 0.

Review

Auxiliary Equations

Given a linear homogeneous constant-coefficient differential equation

an d

n (^) y dxn^ +^ an−^1

dn−^1 y dxn−^1 +^ ...a^1

dy dx +^ a^0 y^ = 0,

the Auxiliary Equation is

anmn^ + an− 1 mn−^1 + ...a 1 m + a 0 = 0.

The Auxiliary Equation determines the general solution.

Review

General Solution from the Auxiliary Equation

(^1) If m is a root of the auxiliary equation of multiplicity k then emx^ , xemx^ , x^2 emx^ , ... , xk−^1 emx^ are linearly independent solutions. (^2) If (α + iβ) and (α + iβ) are a roots of the auxiliary equation of multiplicity k then eαx^ cos(βx), xeαx^ cos(βx), ... , xk−^1 eαx^ cos(βx) and eαx^ sin(βx), xeαx^ sin(βx), ... , xk−^1 eαx^ sin(βx) are linearly independent solutions.

Today’s Goals

Today’s Goals

(^1) Learn how to solve nonhomogeneous linear differential equations using the method of Undetermined Coefficients.

Undetermined Coefficients

The Method of Undetermined Coefficients

Given a nonhomogeneous differential equation

any (n)^ + an− 1 y (n−1)^ + ...a 1 y ′^ + a 0 y = g (x) where an, an− 1 , ..., a 0 are constants. (^1) Step 1: Solve the associated homogeneous equation.

Undetermined Coefficients

The Method of Undetermined Coefficients

Given a nonhomogeneous differential equation

any (n)^ + an− 1 y (n−1)^ + ...a 1 y ′^ + a 0 y = g (x) where an, an− 1 , ..., a 0 are constants. (^1) Step 1: Solve the associated homogeneous equation. (^2) Step 2: Find a particular solution by analyzing g(x) and making an educated guess.

Undetermined Coefficients

Guessing Particular Solutions

g(x) Guess constant

Undetermined Coefficients

Guessing Particular Solutions

g(x) Guess constant A

Undetermined Coefficients

Guessing Particular Solutions

g(x) Guess constant A 3 x^2 − 2 Ax^2 + Bx + C

Undetermined Coefficients

Guessing Particular Solutions

g(x) Guess constant A 3 x^2 − 2 Ax^2 + Bx + C Polynomial of degree n

Undetermined Coefficients

Guessing Particular Solutions

g(x) Guess constant A 3 x^2 − 2 Ax^2 + Bx + C Polynomial of degree n Anxn^ + An− 1 xn−^1 + ... + A 0 cos(4x)

Undetermined Coefficients

Guessing Particular Solutions

g(x) Guess constant A 3 x^2 − 2 Ax^2 + Bx + C Polynomial of degree n Anxn^ + An− 1 xn−^1 + ... + A 0 cos(4x) Acos(4x) + Bsin(4x)