   Chapter 17.2, Problem 25E

Chapter
Section
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,

y3y+2y=11+ex (1)

Consider the auxiliary equation is,

r23r+2=0 (2)

Roots of equation (2) are,

r=(3)±(3)24(1)(2)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=3±12=1and2

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 y1y2y2y1 .

y1y2y2y1=exd(e2x)dxe2xd(ex)dx=ex(2)(e2x)(e2x)(ex)(1)=2e3xe3x=e3x

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 11+ex for G(x) , e2x for y2 , and e3x for y1y2y2y1 ,

u1=11+ex(e2x)e3x=ex1+ex

Integrate on both sides of the equation.

u1=ex1+exdxu1(x)=ln(1+ex)

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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