Chapter 17.1, Problem 21E

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.21. y" – 6y' + 10y = 0, y(0) = 2, y'(0) = 3

To determine

To solve: The initial-value problem for differential equation y6y+10y=0 , y(0)=2 , y(0)=3 .

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.

yâ€³âˆ’6yâ€²+10y=0 (5)

Compare equation (1) and (5).

a=1b=âˆ’6c=10

Find the auxiliary equation.

Substitute 1 for a , âˆ’6 for b and 10 for c in equation (2),

(1)r2+(âˆ’6)r+(10)=0r2âˆ’6r+10=0

Simplify equation as follows.

r=âˆ’(âˆ’6)Â±(âˆ’6)2âˆ’4(1)(10)2(1)=6Â±36âˆ’402=6Â±âˆ’42

r=3Â±i (6)

Compare equation (3) and (6).

Î±=3Î²=1

Find the general solution of yâ€³âˆ’6yâ€²+10y=0 using equation (4).

Substitute 3 for Î± and 1 for Î² in equation (4),

y=e3x(c1cos(1)x+c2sin(1)x)

y=e3x(c1cosx+c2sinx) (7)

Modify equation (7) as follows

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