   Chapter 17.2, Problem 27E

Chapter
Section
Textbook Problem

Solve the differential equation using the method of variation of parameters.27. y ″ − 2 y ′ + y = e x 1 + x 2

To determine

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

Explanation

Given data:

The differential equation is,

y2y+y=ex1+x2 (1)

Consider the auxiliary equation.

r22r+1=0 (2)

Roots of equation (2) are,

r=(2)±(2)24(1)(1)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=22=1

Write the expression for the complementary solution of the one real root.

yc(x)=c1erx+c2xerx (3)

Substitute 1 for r in equation (3),

yc(x)=c1e1x+c2xe1x

yc(x)=c1ex+c2xex (4)

From equation (4), set y1=ex and y2=xex .

Calculate y1y2y2y1 .

y1y2y2y1=exd(xex)dxxexd(ex)dx=ex(xex+ex(1))xexex=ex(x+1)exxexex=xexex+exexxexex

y1y2y2y1=exex=e2x

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 ex1+x2 for G(x) , xex for y2 , and e2x for y1y2y2y1 ,

u1=ex1+x2(xex)e2x=x1+x2

Integrate on both sides of the equation.

u1=x1+x2dxu1(x)=12ln(1+x2)

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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