   # (a) For what values of r does the function y = e rx satisfy the differential equation 2 y ″ + y ′ − y = 0? (b) If r 1 and r 2 are the values of r that you found in part (a), show that every member of the family of functions y = a e r 1 x + b e r 2 x is also a solution. ### Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550

#### Solutions

Chapter
Section ### Calculus: Early Transcendentals

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285741550
Chapter 9.1, Problem 3E
Textbook Problem
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## (a) For what values of r does the function y = erx satisfy the differential equation 2y″ + y′ − y = 0?(b) If r1 and r2 are the values of r that you found in part (a), show that every member of the family of functions y = a e r 1 x + b e r 2 x is also a solution.

(a)

To determine

To find: The values of r when the function y=erx satisfy the differential equation 2y+yy=0 .

### Explanation of Solution

Given:

The function is y=erx .

The differential equation is 2y+yy=0 .

Calculation:

Obtain the values of r when the function y=erx satisfy the differential equation 2y+yy=0 .

Differentiate the function y with respect to x,

y=ddx(y)=ddx(erx)=rex

Thus, y=rerx .

Differentiate the function y with respect to x,

y=ddx(y)=ddx(rerx)=r2erx

Thus, y=r2erx .

Since the function y=erx satisfy the differential equation 2y+yy=0

Substitute the values of y,y and y in the differential equation 2y+yy=0 to compute the values of r as follows,

2y+yy=0

(b)

To determine

To show: Every member of the family of functions y=aer1x+ber2x is a solution of the differential equation 2y+yy=0

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