   Chapter 17, Problem 13RE

Chapter
Section
Textbook Problem

Solve the initial-value problem.13. y" – 5y' + 4y = 0, y(0) = 0, y'(0) = 1

To determine

To solve: The initial-value problem for differential equation y5y+4y=0 , y(0)=0 , y(0)=1 .

Explanation

Given data:

The differential equation is,

y5y+4y=0 (1)

Formula used:

Write the expression for a second order differential equation.

ay+by+cy=0 (2)

Write the expression for an auxiliary equation.

ar2+br+c=0 (3)

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

y(x)=c1er1x+c2er2x (4)

Here,

r1 and r2 are the roots of the auxiliary equation.

Write the expression to find the roots of quadratic equation.

r=b±b24ac2a (5)

Compare equation (1) and (2).

a=1b=5c=4

Substitute 1 for a, –5 for b, and 4 for c in equation (3),

(1)r2+(5)r+(4)=0r25r+4=0

Find the roots of auxiliary equation using equation (5).

Substitute 1 for a, –5 for b, and 4 for c in equation (5),

r=(5)±(5)24(1)(4)2(1)=5±25162=5±92=5±32

Here, the value of r1 is 1 and the value of r2 is 4.

Substitute 1 for r1 and 4 for r2 in equation (4),

y(x)=c1ex+c2e4x (6)

Substitute 0 for x in equation (6) and find the value of y(0)

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