Chapter 17.2, Problem 24E

### 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 using the method of variation of parameters.24. y" + y = sec3x, 0 < x < π/2

To determine

To solve: The differential equation by using method of variation of parameters.

Explanation

Given data:

The differential equation is,

yâ€³+y=sec3x,â€‰0<x<Ï€2 (1)

Consider the auxiliary equation.

r2+1=0 (2)

Roots of equation (2) are,

r=âˆ’0Â±(0)2âˆ’4(1)(1)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€‰}=Â±2i2=Â±i

Write the expression for the complementary solution for the complex roots,

yc(x)=eÎ±x(c1cosÎ²x+c2sinÎ²x)

Substitute 0 for Î± and 1 for Î² ,

yc(x)=e0x(c1cos1x+c2sin1x)

yc(x)=c1cosx+c2sinx (3)

Set y1=sinx and y2=cosx .

Calculate y1yâ€²2âˆ’y2yâ€²1 .

y1yâ€²2âˆ’y2yâ€²1=sinxd(cosx)dxâˆ’cosxd(sinx)dx=sinx(âˆ’sinx)âˆ’cosxcosx=âˆ’sin2xâˆ’cos2x=âˆ’(sin2x+cos2x)â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰{âˆµsin2x+cos2x=1}

y1yâ€²2âˆ’y2yâ€²1=âˆ’1

Write the expression to find the arbitrary function uâ€²1 .

uâ€²1=âˆ’G(x)y2y1yâ€²2âˆ’y2yâ€²1

Here,

G(x) is the expression for R.H.S of differential equation in (1),

Substitute sec3x for G(x) , cosx for y2 , and âˆ’1 for y1yâ€²2âˆ’y2yâ€²1 ,

uâ€²1=âˆ’sec3xcosxâˆ’1=âˆ’sec3x1secxâˆ’1â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰{âˆµcosx=1secx}=sec2x

Integrate on both sides of the equation.

âˆ«uâ€²1=âˆ«sec2xdxu1(x)=tanx

Write the expression to find the arbitrary function uâ€²2 .

uâ€²2=G(x)y1y1yâ€²2âˆ’y2yâ€²1

Here,

G(x) is the expression for R

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started