   Chapter 17.1, Problem 22E

Chapter
Section
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.

ay+by+cy=0 (1)

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression for general solution of ay+by+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.

4y20y+25y=0 (4)

Compare equation (1) and (4).

a=4b=20c=25

Find the auxiliary equation.

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

(4)r2+(20)r+(25)=04r220r+25=0(2r5)2=02r5=0

Simplify the equation as follows.

2r=5r=52

Find the general solution of 4y20y+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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