Chapter 17.2, Problem 25E

### 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.25. y ″ − 3 y ′ + 2 y = 1 1 + e − x

To determine

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

Explanation

Given data:

The differential equation is,

yâ€³âˆ’3yâ€²+2y=11+eâˆ’x (1)

Consider the auxiliary equation is,

r2âˆ’3r+2=0 (2)

Roots of equation (2) are,

r=âˆ’(âˆ’3)Â±(âˆ’3)2âˆ’4(1)(2)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€‰}=3Â±12=1â€‰andâ€‰2

Write the expression for the complementary solution of two real roots.

yc(x)=c1er1x+c2er2x (3)

Substitute 1 for r1 and 2 for r2 in equation (3),

yc(x)=c1e1x+c2e2x

yc(x)=c1ex+c2e2x (4)

From equation (4), set y1=ex and y2=e2x .

Calculate y1yâ€²2âˆ’y2yâ€²1 .

y1yâ€²2âˆ’y2yâ€²1=exd(e2x)dxâˆ’e2xd(ex)dx=ex(2)(e2x)âˆ’(e2x)(ex)(1)=2e3xâˆ’e3x=e3x

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 11+eâˆ’x for G(x) , e2x for y2 , and e3x for y1yâ€²2âˆ’y2yâ€²1 ,

uâ€²1=âˆ’11+eâˆ’x(e2x)e3x=âˆ’eâˆ’x1+eâˆ’x

Integrate on both sides of the equation.

âˆ«uâ€²1=âˆ’âˆ«eâˆ’x1+eâˆ’xdxu1(x)=ln(1+eâˆ’x)

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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