   Chapter 17.1, Problem 29E

Chapter
Section
Textbook Problem

Solve the boundary-value problem, if possible.29. y" = y', y(0) = 1, y(1) = 2

To determine

To solve: The boundary-value problem for differential equation y=y , y(0)=1 , y(1)=2 .

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 two distinct real roots (r) .

y=c1er1x+c2er2x (3)

Here,

r1 and r2 is the root of auxiliary equation.

Consider the differential equation as follows.

y=y

yy=0 (4)

Compare equation (1) and (4).

a=1b=1c=0

Find the auxiliary equation.

Substitute 1 for a , 1 for b , and 0 for c in equation (2),

(1)r2+(1)r+(0)=0r2r=0r(r1)=0

Solve for r .

r=0r=1

Consider the value of r1 and r2 as follows.

r1=0r2=1

Find the general solution of y=y using equation (3).

Substitute 0 for r1 and 1 for r2 in equation (3),

y=c1e(0)x+c2e(1)x

y=c1+c2ex (5)

Modify equation (5) as follows.

y(x)=c1+c2ex (6)

Find the value of y(0) .

Substitute 0 for x in equation (6),

y(0)=c1+c2e0=c1+c2

Substitute 1 for y(0) ,

1=c1+c2 (7)

Find the value of y(1)

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