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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.

10. d 2 y d x 2 + y = csc x , 0 < x < π / 2

To determine

To solve: The differential equation.

Explanation

Given data:

The differential equation is,

d2ydx2+y=cscx

y+y=cscx (1)

The method of variation of parameters can be used to solve the differential equation in equation (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}=±i22=±i

Write the expression for the complementary solution of two complex roots r=α±iβ ,

yc(x)=eαx(c1cosβx+c2sinβx) (3)

Substitute 0 for α and 1 for β in equation (3),

yc(x)=e0x(c1cos1x+c2sin1x)

yc(x)=c1cosx+c2sinx (4)

From equation (4), set y1=cosx and y2=sinx .

Calculate y1y2y2y1 .

y1y2y2y1=cosxd(sinx)dxsinxd(cosx)dx=cosx(cosx)sinx(sinx)=cos2x+sin2x=1{cos2x+sin2x=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 cscx for G(x) , sinx for y2 , and 1 for y1y2y2y1 ,

u1=cscx(sinx)1=cscx(sinx)=1sinx(sinx)=1{cscx=1sinx}

Integrate on both sides of the equation

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Chapter 17 Solutions

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Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

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