   Chapter 17.1, Problem 23E

Chapter
Section
Textbook Problem

Solve the initial-value problem.23. y" – y' – 12y = 0, y(1) = 0, y'(1) = 1

To determine

To solve: The initial-value problem for differential equation yy12y=0 , y(1)=0 , y(1)=1 .

Explanation

Formula used:

Write the expression for differential equation.

ay+by+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for general solution of ay+by+cy=0 with two distinct real roots.

y=c1er1x+c2er2x (3)

Here,

r1 and r2 are the roots of auxiliary equation.

Write the required differential formulae to evaluate the differential equation.

ddxenx=nenx

Consider the differential equation as follows.

yy12y=0 (4)

Compare equation (1) and (4).

a=1b=1c=12

Find the auxiliary equation.

Substitute 1 for a , 1 for b and 12 for c in equation (2),

(1)r2+(1)r+(12)=0r2r12=0(r4)(r+3)=0

Simplify the equation as follows.

r4=0r=4r+3=0r=3

Consider the value of r1 and r2 as follows.

r1=4r2=3

Find the general solution of yy12y=0 using equation (3).

Substitute 4 for r1 and 3 for r2 in equation (3),

y=c1e4x+c2e3x (5)

Modify equation (5) as follows.

y(x)=c1e4x+c2e3x (6)

Find the value of y(1) .

Substitute 1 for x in equation (6),

y(1)=c1e4(1)+c2e3(1)=c1e4+c2e3

Substitute 0 for y(1) ,

0=c1e4+c2e3

Re-arrange equation as follows

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