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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Section
BuyFindarrow_forward

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:

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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Chapter 17 Solutions

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Sect-17.1 P-11ESect-17.1 P-12ESect-17.1 P-13ESect-17.1 P-14ESect-17.1 P-15ESect-17.1 P-16ESect-17.1 P-17ESect-17.1 P-18ESect-17.1 P-19ESect-17.1 P-20ESect-17.1 P-21ESect-17.1 P-22ESect-17.1 P-23ESect-17.1 P-24ESect-17.1 P-25ESect-17.1 P-26ESect-17.1 P-27ESect-17.1 P-28ESect-17.1 P-29ESect-17.1 P-30ESect-17.1 P-31ESect-17.1 P-32ESect-17.1 P-33ESect-17.1 P-34ESect-17.2 P-1ESect-17.2 P-2ESect-17.2 P-3ESect-17.2 P-4ESect-17.2 P-5ESect-17.2 P-6ESect-17.2 P-7ESect-17.2 P-8ESect-17.2 P-9ESect-17.2 P-10ESect-17.2 P-11ESect-17.2 P-12ESect-17.2 P-13ESect-17.2 P-14ESect-17.2 P-15ESect-17.2 P-16ESect-17.2 P-17ESect-17.2 P-18ESect-17.2 P-19ESect-17.2 P-20ESect-17.2 P-21ESect-17.2 P-22ESect-17.2 P-23ESect-17.2 P-24ESect-17.2 P-25ESect-17.2 P-26ESect-17.2 P-27ESect-17.2 P-28ESect-17.3 P-1ESect-17.3 P-2ESect-17.3 P-3ESect-17.3 P-4ESect-17.3 P-5ESect-17.3 P-6ESect-17.3 P-7ESect-17.3 P-8ESect-17.3 P-9ESect-17.3 P-10ESect-17.3 P-11ESect-17.3 P-12ESect-17.3 P-13ESect-17.3 P-14ESect-17.3 P-15ESect-17.3 P-16ESect-17.3 P-17ESect-17.3 P-18ESect-17.4 P-1ESect-17.4 P-2ESect-17.4 P-3ESect-17.4 P-4ESect-17.4 P-5ESect-17.4 P-6ESect-17.4 P-7ESect-17.4 P-8ESect-17.4 P-9ESect-17.4 P-10ESect-17.4 P-11ESect-17.4 P-12ECh-17 P-1RCCCh-17 P-2RCCCh-17 P-3RCCCh-17 P-4RCCCh-17 P-5RCCCh-17 P-1RQCh-17 P-2RQCh-17 P-3RQCh-17 P-4RQCh-17 P-1RECh-17 P-2RECh-17 P-3RECh-17 P-4RECh-17 P-5RECh-17 P-6RECh-17 P-7RECh-17 P-8RECh-17 P-9RECh-17 P-10RECh-17 P-11RECh-17 P-12RECh-17 P-13RECh-17 P-14RECh-17 P-15RECh-17 P-16RECh-17 P-17RECh-17 P-18RECh-17 P-19RECh-17 P-20RECh-17 P-21RE

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