Chapter 17, Problem 4RQ

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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:

yâ€³âˆ’y=ex

Formula used:

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

ayâ€³+byâ€²+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.

yâ€³âˆ’y=ex (4)

Compare equations (1) and (4).

a=1b=0c=âˆ’1G(x)=ex

Write the expression for auxiliary equation.

ar2+br+c=0

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

(1)r2+(0)r+(âˆ’1)=0r2âˆ’1=0r2=1r=Â±1

Consider the value of Â  r1 and r2 as follows

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