   Chapter 17, Problem 6RE

Chapter
Section
Textbook Problem

Solve the differential equation.6.  d 2 y d x 2 + d y d x − 2 y = x 2

To determine

To solve: The differential equation of d2ydx2+dydx2y=x2 .

Explanation

Given data:

The differential equation is,

d2ydx2+dydx2y=x2

y+y2y=x2 (1)

Formula used:

Write the expression for differential equation.

ay+by+cy=0 (2)

Write the expression for auxiliary equation.

ar2+br+c=0 (3)

Write the expression for general solution of ay+by+cy=0 with two distinct real roots.

yc(x)=c1er1x+c2er2x (4)

Here,

r1 and r2 are the roots of auxiliary equation.

Write the expression for the particular solution yp(x) .

yp(x)=Ax2+Bx+C (5)

Write the expression to find the roots of quadratic equation.

r=b±b24ac2a (6)

Compare equations (1) and (2).

a=1b=1c=2

Substitute 1 for a, 1 for b, and –2 for c in equation (3),

(1)r2+(1)r+(2)=0r2+r2=0

Find the roots of auxiliary equation using equation (6).

Substitute 1 for a, 1 for b, and –2 for c in equation (6),

r=(1)±(1)24(1)(2)2(1)=1±1+82=1±92=1±32

Consider the value of r1 and r2 as follows.

r1=1+32=22=1

r2=132=2

Substitute 1 for r1 and –2 for r2 in equation (4),

yc(x)=c1ex+c2e2x (7)

Differentiate equation (5) with respect to x,

yp(x)=ddx(Ax2+Bx+C)

yp(x)=2Ax+B (8)

Differentiate equation (8) with respect to x,

yp(x)=ddx(2Ax+B)

yp(x)=2A (9)

Substitute equations (5), (8) and (9) in equation (1),

2Ax+B+2A2(Ax2+Bx+C)=x2

2A+2Ax+B2Ax22Bx2C=x2 (10)

Substitute 0 for x in equation (10),

2A+2A(0)+B2A(0)

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