   Chapter 17.2, Problem 2E

Chapter
Section
Textbook Problem

Solve the differential equation or initial-value problem using the method of undetermined coefficients.2. y" – 3y' = sin 2x

To determine

To solve: The differential equation by the method of undetermined coefficients.

Explanation

Given data:

The differential equation is,

y3y=sin2x (1)

Consider the auxiliary equation is,

r23r=0 (2)

Roots of equation (2) are,

r=(3)±(3)24(1)(0)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=3±32=0and3

Write the expression for the complementary solution for two real roots,

yc(x)=c1er1x+c2er2x

Substitute 0 for r1 and 3 for r2 ,

yc(x)=c1e0x+c2e3x

yc(x)=c1+c2e3x (3)

The Right hand side (RHS) of a differential equation contains only sine function. The particular solution for this case can be expressed as follows.

yp(x)=Acos2x+Bsin2x (4)

Differentiate equation (4) with respect to x,

yp(x)=ddx(Acos2x+Bsin2x)

yp(x)=2Asin2x+2Bcos2x (5)

Differentiate equation (5) with respect to x,

yp(x)=ddx(2Asin2x+2Bcos2x)

yp(x)=4Acos2x4Bsin2x (6)

Substitute equations (5) and (6) in (1),

4Acos2x4Bsin2x3(2Asin2x+2Bcos2x)=sin2x

(4A6B)cos2x+(6A4B)sin2x=sin2x (7)

Substitute 0 for x in equation (7),

(4A6B)cos2(0)+(6A4B

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