   Chapter 17.1, Problem 34E

Chapter
Section
Textbook Problem

If a, b, and c are all positive constants and y(x) is a solution of the differential equation ay" + by' + cy = 0, show that limx→∞ y(x) = 0.

To determine

To show: If a , b , and c are all positive constant then the general solution of second –order differential equation results in limxy(x)=0 .

Explanation

Given data:

ay+by+cy=0 (1)

Formula used:

Write the expression for auxiliary equation.

ar2+br+c=0 (2)

Write the expression to find the roots of auxiliary equation.

r=b±b24ac2a (3)

Write the expression for general solution with real and distinct roots.

y=c1er1x+c2er2x (4)

Write the expression for general solution with real and same roots.

y=c1erx+c2xerx (5)

Write the expression for general solution with complex roots.

y=eαx(c1cosβx+c2sinβx) (6)

Case I:

Consider the value of b24ac>0 in equation (3).

Consider root r consists of two roots represented as follows.

r1=b+b24ac2ar2=bb24ac2a

Substitute b+b24ac2a for r1 and bb24ac2a for r2 in equation (4),

y=c1e(b+b24ac2a)x+c2e(bb24ac2a)x (7)

Modify equation (7) as follows.

y(x)=c1e(b+b24ac2a)x+c2e(bb24ac2a)x (8)

Since, a , b , and c are all positive constant, the value of r1 and r2 has negative value.

Modify equation (8) as follows.

y(x)=c1er1x+c2er2x

Find the value of limxy(x) .

limxy(x)=c1er1()+c2er2()=c1(1e)+c2(1e)=c1(1)+c2(1)=0

Case II:

Consider the value of b24ac=0 in equation (3).

In this case, the roots of auxiliary equation are real and equal.

Substitute 0 for b24ac in equation (3),

r=b±02a=b2a w

Substitute b2a for r in equation (5),

y=c1e(b2a)x+c2xe(b2a)x (9)

Modify equation (9) as follows

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