   Chapter 17, Problem 9RE

Chapter
Section
Textbook Problem

Solve the differential equation.9. d 2 y d x 2 − d y d x − 6 y = 1 + e − 2 x

To determine

To solve: The differential equation.

Explanation

Given data:

The differential equation is,

d2ydx2dydx6y=1+e2x

yy6y=1+e2x (1)

Consider the auxiliary equation.

r2r6=0 (2)

Roots of equation (2) are,

r=(1)±(1)24(1)(6)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=1±52=2and3

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

yc(x)=c1er1x+c2er2x

Substitute 2 for r1 and 3 for r2 ,

yc(x)=c1e2x+c2e3x (3)

Re-arrange equation (1) by neglecting e2x ,

yy6y=1 (4)

Find the Particular solution for equation (4),

The Right hand side (RHS) of a differential equation contains only numerical value. Then, the particular solution for this case can be expressed as follows.

yp1(x)=A (5)

Differentiate equation (5) with respect to x.

yp1(x)=ddx(0)

yp1(x)=0 (6)

Differentiate equation (6) with respect to x.

yp1(x)=ddx(0)

yp1(x)=0 (7)

Substitute equations (5), (6), and (7) in equation (4),

006A=1A=16

Re-arrange equation (1) by neglecting 1,

yy6y=e2x (8)

Find the Particular solution for equation (8)

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