Chapter 17.1, Problem 34E

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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:

ayâ€³+byâ€²+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=âˆ’bÂ±b2âˆ’4ac2a (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=eÎ±x(c1cosÎ²x+c2sinÎ²x) (6)

Case I:

Consider the value of b2âˆ’4ac>0 in equation (3).

Consider root r consists of two roots represented as follows.

r1=âˆ’b+b2âˆ’4ac2ar2=âˆ’bâˆ’b2âˆ’4ac2a

Substitute âˆ’b+b2âˆ’4ac2a for r1 and âˆ’bâˆ’b2âˆ’4ac2a for r2 in equation (4),

y=c1e(âˆ’b+b2âˆ’4ac2a)x+c2e(âˆ’bâˆ’b2âˆ’4ac2a)x (7)

Modify equation (7) as follows.

y(x)=c1e(âˆ’b+b2âˆ’4ac2a)x+c2e(âˆ’bâˆ’b2âˆ’4ac2a)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)=c1eâˆ’r1x+c2eâˆ’r2x

Find the value of limxâ†’âˆžy(x) .

limxâ†’âˆžy(x)=c1eâˆ’r1(âˆž)+c2eâˆ’r2(âˆž)=c1(1eâˆž)+c2(1eâˆž)=c1(1âˆž)+c2(1âˆž)=0

Case II:

Consider the value of b2âˆ’4ac=0 in equation (3).

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

Substitute 0 for b2âˆ’4ac in equation (3),

r=âˆ’bÂ±02a=âˆ’b2a w

Substitute âˆ’b2a for r in equation (5),

y=c1e(âˆ’b2a)x+c2xe(âˆ’b2a)x (9)

Modify equation (9) as follows

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