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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

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,

y+y=sec3x,0<x<π2 (1)

Consider the auxiliary equation.

r2+1=0 (2)

Roots of equation (2) are,

r=0±(0)24(1)(1)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=±2i2=±i

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

yc(x)=eαx(c1cosβx+c2sinβx)

Substitute 0 for α and 1 for β ,

yc(x)=e0x(c1cos1x+c2sin1x)

yc(x)=c1cosx+c2sinx (3)

Set y1=sinx and y2=cosx .

Calculate y1y2y2y1 .

y1y2y2y1=sinxd(cosx)dxcosxd(sinx)dx=sinx(sinx)cosxcosx=sin2xcos2x=(sin2x+cos2x){sin2x+cos2x=1}

y1y2y2y1=1

Write the expression to find the arbitrary function u1 .

u1=G(x)y2y1y2y2y1

Here,

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

Substitute sec3x for G(x) , cosx for y2 , and 1 for y1y2y2y1 ,

u1=sec3xcosx1=sec3x1secx1{cosx=1secx}=sec2x

Integrate on both sides of the equation.

u1=sec2xdxu1(x)=tanx

Write the expression to find the arbitrary function u2 .

u2=G(x)y1y1y2y2y1

Here,

G(x) is the expression for R

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