Chapter 17.2, Problem 2E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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,

yâ€³âˆ’3yâ€²=sin2x (1)

Consider the auxiliary equation is,

r2âˆ’3r=0 (2)

Roots of equation (2) are,

r=âˆ’(âˆ’3)Â±(âˆ’3)2âˆ’4(1)(0)2(1)â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰{âˆµr=âˆ’bÂ±b2âˆ’4ac2aforâ€‰theâ€‰equationâ€‰ofar2+br+c=0â€‰â€‰}=3Â±32=0â€‰andâ€‰3

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,

yâ€²p(x)=ddx(Acos2x+Bsin2x)

yâ€²p(x)=âˆ’2Asin2x+2Bcos2x (5)

Differentiate equation (5) with respect to x,

yâ€³p(x)=ddx(âˆ’2Asin2x+2Bcos2x)

yâ€³p(x)=âˆ’4Acos2xâˆ’4Bsin2x (6)

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

âˆ’4Acos2xâˆ’4Bsin2xâˆ’3(âˆ’2Asin2x+2Bcos2x)=sin2x

(âˆ’4Aâˆ’6B)cos2x+(6Aâˆ’4B)sin2x=sin2x (7)

Substitute 0 for x in equation (7),

(âˆ’4Aâˆ’6B)cos2(0)+(6Aâˆ’4B

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