Chapter 17.1, Problem 16E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Graph the two basic solutions along with several other solutions of the differential equation. What features do the solutions have in common?16. 2 d 2 y d x 2 + d y d x   −   y   =   0

To determine

To graph: The two basic solutions along with several other solutions of the differential equation for 2d2ydx2+dydxy=0 .

Explanation

Formula used:

Write the expression for differential equation.

ayâ€³+byâ€²+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for general solution of ayâ€³+byâ€²+cy=0 with two distinct real roots.

y=c1er1x+c2er2x (3)

Consider the differential equation as follows.

2d2ydx2+dydxâˆ’y=0 (4)

Compare equation (1) and (4).

a=2b=1c=âˆ’1

Find the auxiliary equation.

Substitute 2 for a , 1 for b and âˆ’1 for c in equation (2),

(2)r2+(1)r+(âˆ’1)=02r2+râˆ’1=0(2râˆ’1)(r+1)=0

Solve for r .

2râˆ’1=0r=12r+1=0r=âˆ’1

Consider the value of r1 and r2 as follows,

r1=12r2=âˆ’1

Substitute 12 for r1 and âˆ’1 for r2 in equation (3),

y=c1e12x+c2eâˆ’x (5)

Here,

c1 and c2 are constant value.

Consider the value of c1 and c2 as 1 .

Substitute 1 for c1 and 1 for c2 in equation (5),

y=ex2+eâˆ’x (6)

Equation (6) consists two functions f(x) and g(x) as follows.

f(x)=ex2 (7)

g(x)=eâˆ’x (8)

Here,

f(x) and g(x) are two basic solutions.

Consider the constant value for c1 and c2 as follows

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