   Chapter 17, Problem 10RE

Chapter
Section
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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