   Chapter 17.1, Problem 30E

Chapter
Section
Textbook Problem

Solve the boundary-value problem, if possible.30. 4y" − 4y' + y = 0, y(0) = 4, y(2) = 0

To determine

To solve: The boundary-value problem for differential equation 4y4y+y=0 , y(0)=4 , y(2)=0 .

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 only one real root r .

y=c1erx+c2xerx (3)

Here,

r is the root of auxiliary equation.

Consider the differential equation as follows.

4y4y+y=0 (4)

Compare equation (1) and (4).

a=4b=4c=1

Find the auxiliary equation.

Substitute 4 for a , 4 for b and 1 for c in equation (2),

(4)r2+(4)r+(1)=04r24r+1=0(2r1)2=02r1=0

Solve for r .

2r=1r=12

Find the general solution of 4y4y+y=0 using equation (3).

Substitute 12 for r in equation (3),

y=c1e12x+c2xe12x

y=c1ex2+c2xex2 (5)

Modify equation (5) as follows.

y(x)=c1ex2+c2xex2 (6)

Find the value of y(0)

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