Chapter 17.2, Problem 27E

### 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.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,

yâ€³âˆ’2yâ€²+y=ex1+x2 (1)

Consider the auxiliary equation.

r2âˆ’2r+1=0 (2)

Roots of equation (2) are,

r=âˆ’(âˆ’2)Â±(âˆ’2)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€‰}=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 y1yâ€²2âˆ’y2yâ€²1 .

y1yâ€²2âˆ’y2yâ€²1=exd(xex)dxâˆ’xexd(ex)dx=ex(xex+ex(1))âˆ’xexex=ex(x+1)exâˆ’xexex=xexex+exexâˆ’xexex

y1yâ€²2âˆ’y2yâ€²1=exex=e2x

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 ex1+x2 for G(x) , xex for y2 , and e2x for y1yâ€²2âˆ’y2yâ€²1 ,

uâ€²1=âˆ’ex1+x2(xex)e2x=âˆ’x1+x2

Integrate on both sides of the equation.

âˆ«uâ€²1=âˆ’âˆ«x1+x2dxu1(x)=âˆ’12ln(1+x2)

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

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