   Chapter 17.1, Problem 18E

Chapter
Section
Textbook Problem

Solve the initial-value problem.18. y" - 2y' – 3y = 0, y(0) = 2, y'(0) = 2

To determine

To solve: The initial-value problem for differential equation y2y3y=0 , y(0)=2 , y(0)=2 .

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.

y2y3y=0 (4)

Compare equation (1) and (4).

a=1b=2c=3

Find the auxiliary equation.

Substitute 1 for a , 2 for b and 3 for c in equation (2),

(1)r2+(2)r+(3)=0r22r3=0(r3)(r+1)=0

Simplify the equation as follows.

r3=0r=3r+1=0r=1

Find the general solution of y2y3y=0 using equation (3).

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

y=c1e3x+c2e(1)x

y=c1e3x+c2ex (5)

Modify equation (5) as follows.

y(x)=c1e3x+c2ex (6)

Find the value of y(0) .

Substitute 0 for x in equation (6),

y(0)=c1e3(0)+c2e(0)=c1e0+c2e0=c1+c2

Substitute 2 for y(0) ,

2=c1+c2

Re-arrange equation as follows

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