   Chapter 17.2, Problem 28E

Chapter
Section
Textbook Problem

Solve the differential equation using the method of variation of parameters.28. y ″ + 4 y ′ + 4 y = e − 2 x x 3

To determine

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

Explanation

Given data:

The differential equation is,

y+4y+4y=e2xx3 (1)

Consider the auxiliary equation is,

r2+4r+4=0 (2)

Roots of equation (2) are,

r=(4)±(4)24(1)(4)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=42=2

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

yc(x)=c1erx+c2xerx (3)

Substitute 2 for r in equation (3),

yc(x)=c1e2x+c2xe2x (4)

From equation (4), set y1=e2x and y2=xe2x .

Calculate y1y2y2y1 .

y1y2y2y1=e2xd(xe2x)dxxe2xd(e2x)dx=e2x(xe2x(2)+e2x(1))xe2xe2x(2)=e2x(2x+1)e2x+2xe2xe2x=2xe2xe2x+e2xe2x+2xe2xe2x

Simplify the equation.

y1y2y2y1=e2xe2x=e4x

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 e2xx3 for G(x) , xe2x for y2 , and e4x for y1y2y2y1 ,

u1=e2xx3(xe2x)e4x=e2x(xe2x)x3e4x=1x2

Integrate on both sides of the equation

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