   Chapter 17.2, Problem 1E

Chapter
Section
Textbook Problem

Solve the differential equation or initial-value problem using the method of undetermined coefficients.1. y" + 2y' – 8y = 1 – 2x2

To determine

To solve: The differential equation by the method of undetermined coefficients.

Explanation

Given data:

The differential equation is,

y+2y8y=12x2 (1)

Consider the auxiliary equation.

r2+2r8=0 (2)

Roots of equation (2) are,

r=(2)±(2)24(1)(8)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=2±62=2and4

Write the expression for the complementary solution for two real roots,

yc(x)=c1er1x+c2er2x

Substitute 2 for r1 and 4 for r2 ,

yc(x)=c1e2x+c2e4x (3)

The Right hand side (RHS) of a differential equation contains 2nd order polynomial and numerical value. The particular solution for this case can be expressed as follows.

yp(x)=Ax2+Bx+C (4)

Differentiate equation (4) with respect to x,

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

yp(x)=2Ax+B (5)

Differentiate equation (5) with respect to x,

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

yp(x)=2A (6)

Substitute equations (4), (5), and (6) in equation (1),

2A+2(2Ax+B)8(A</

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