Chapter 17.1, Problem 22E

### 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.22. 4y" – 20y' + 25y = 0, y(0) = 2, y'(0) = –3

To determine

To solve: The initial-value problem for differential equation 4y20y+25y=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 general solution of ayâ€³+byâ€²+cy=0 with same real roots.

y=c1erx+c2xerx (3)

Here,

r is the root of auxiliary equation.

Write the required differential formulae to evaluate the differential equation.

ddxenx=nenx

Consider the differential equation as follows.

4yâ€³âˆ’20yâ€²+25y=0 (4)

Compare equation (1) and (4).

a=4b=âˆ’20c=25

Find the auxiliary equation.

Substitute 4 for a , âˆ’20 for b and 25 for c in equation (2),

(4)r2+(âˆ’20)r+(25)=04r2âˆ’20r+25=0(2râˆ’5)2=02râˆ’5=0

Simplify the equation as follows.

2r=5r=52

Find the general solution of 4yâ€³âˆ’20yâ€²+25y=0 using equation (3).

Substitute 52 for r in equation (3),

y=c1e52x+c2xe52x (5)

Modify equation (5) as follows.

y(x)=c1e52x+c2xe52x (6)

Find the value of y(0)

Substitute 0 for x in equation (6),

y(0)=c1e52(0)+c2(0)e52(0)=c1e0+0=c1

Substitute 2 for y(0) ,

2=c1

Differentiate equation (6) with respect to x

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