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Chapter 17.1, Problem 27E
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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Solve the boundary-value problem, if possible.27. y" + 4y' + 4y = 0, y(0) = 2, y(1) = 0

To determine

To solve: The boundary-value problem for differential equation y+4y+4y=0 , y(0)=2 , y(1)=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 same real roots r .

y=c1erx+c2xerx (3)

Here,

r is the root of auxiliary equation.

Consider the differential equation as follows.

yâ€³+4yâ€²+4y=0 (4)

Compare equation (1) and (4).

a=1b=4c=4

Find the auxiliary equation.

Substitute 1 for a , 4 for b and 4 for c in equation (2),

(1)r2+(4)r+(4)=0r2+4r+4=0(r+2)2=0r+2=0

Simplify the equation as follows.

r=âˆ’2

Find the general solution of yâ€³+4yâ€²+4y=0 using equation (3).

Substitute âˆ’2 for r in equation (3),

y=c1eâˆ’2x+c2xeâˆ’2x (5)

Modify equation (5) as follows.

y(x)=c1eâˆ’2x+c2xeâˆ’2x (6)

Find the value of y(0)

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