   Chapter 17, Problem 4RQ

Chapter
Section
Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.The equation y" − y = ex has a particular solution of the formyρ = Aex

To determine

Whether the statement “The equation yy=ex has a particular solution of the form yp=Aex ” is true or false.

Explanation

Given data:

yy=ex

Formula used:

Write the general expression for second-order nonhomogeneous linear differential equation as follows.

ay+by+cy=G(x) (1)

Here,

a , b and c are constants, and

G is continuous function.

Write the expression for solution of second order nonlinear homogeneous differential equation.

y(x)=yp(x)+yc(x) (2)

Here,

yp is particular solution, and

yc is general solution of complementary equation.

Write the expression for general solution of complementary equation with two distinct real roots.

yc(x)=c1er1x+c2er2x (3)

Consider the second order differential equation as follows.

yy=ex (4)

Compare equations (1) and (4).

a=1b=0c=1G(x)=ex

Write the expression for auxiliary equation.

ar2+br+c=0

Substitute 1 for a , 0 for b and 1 for c ,

(1)r2+(0)r+(1)=0r21=0r2=1r=±1

Consider the value of   r1 and r2 as follows

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