   Chapter 17.1, Problem 21E

Chapter
Section
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.

ay+by+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for the complex roots.

r=α±iβ (3)

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

y=eαx(c1cosβx+c2sinβx) (4)

Here,

α is the real part of the root, and

β is the imaginary part of the root.

Write the required differential formulae to evaluate the differential equation.

ddxcosx=sinxddxsinx=cosx

Consider the differential equation as follows.

y6y+10y=0 (5)

Compare equation (1) and (5).

a=1b=6c=10

Find the auxiliary equation.

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

(1)r2+(6)r+(10)=0r26r+10=0

Simplify equation as follows.

r=(6)±(6)24(1)(10)2(1)=6±36402=6±42

r=3±i (6)

Compare equation (3) and (6).

α=3β=1

Find the general solution of y6y+10y=0 using equation (4).

Substitute 3 for α and 1 for β 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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