Chapter 17.1, Problem 24E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Solve the initial-value problem.24. 4y" + 4y' + 3y = 0, y(0) = 0, y'(0) = 1

To determine

To solve: The initial-value problem for differential equation 4y+4y+3y=0 , y(0)=0 , y(0)=1 .

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 the complex roots.

r=Î±Â±iÎ² (3)

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

y=eÎ±x(c1cosÎ²x+c2sinÎ²x) (4)

Here,

Î± is the real part of the root, and

Î² is the imaginary part of the root.

Write the required differential formulae to evaluate the differential equation.

ddxcosx=âˆ’sinxddxsinx=cosx

Consider the differential equation as follows.

4yâ€³+4yâ€²+3y=0 (5)

Compare equation (1) and (5).

a=4b=4c=3

Find the auxiliary equation.

Substitute 4 for a , 4 for b and 3 for c in equation (2),

(4)r2+(4)r+(3)=04r2+4r+3=0

Simplify equation as follows.

r=âˆ’(4)Â±(4)2âˆ’4(4)(3)2(4)=âˆ’4Â±16âˆ’488=âˆ’4Â±âˆ’328=âˆ’4Â±4âˆ’28

Simplify equation as follows.

r=âˆ’12Â±i22 (6)

Compare equation (3) and (6).

Î±=âˆ’12Î²=22

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

Substitute âˆ’12 for Î± and 22 for Î² in equation (4),

y=eâˆ’12x(c1cos22x+c2sin22x) (7)

Modify equation (7) as follows.

y(x)=eâˆ’12x(c1cos22x+c2sin22x) (8)

Find the value of y(0) .

Substitute 0 for x in equation (8),

y(0)=eâˆ’12(0)(c1cos22(0)+c2sin22(0))=1(c1+0)=c1

Substitute 0 for y(0) ,

0=c1

Differentiate equation (8) with respect to x

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