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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Solve the differential equation using the method of variation of parameters.

26. y" + 3y' +2y = sin(ex)

To determine

To solve: The differential equation by using method of variation of parameters.

Explanation

Given data:

The differential equation is,

y+3y+2y=sin(ex) (1)

Consider the auxiliary equation is,

r2+3r+2=0 (2)

Roots of equation (2) are,

r=(3)±(3)24(1)(2)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=3±12=1and2

Write the expression for the complementary solution of two real roots.

yc(x)=c1er1x+c2er2x (3)

Substitute 1 for r1 and 2 for r2 in equation (3),

yc(x)=c1e1x+c2e2x

yc(x)=c1ex+c2e2x (4)

From equation (4), set y1=ex and y2=e2x .

Calculate y1y2y2y1 .

y1y2y2y1=exd(e2x)dxe2xd(ex)dx=ex(2)(e2x)(e2x)(ex)(1)=2e3x+e3x=e3x

Write the expression to find the arbitrary function u1 ,

u1=G(x)y2y1y2y2y1

Here,

G(x) is the expression for R.H.S of differential equation in (1),

Substitute sin(ex) for G(x) , e2x for y2 , and e3x for y1y2y2y1 ,

u1=sin(ex)(e2x)e3x=exsinex

Integrate on both sides of the equation.

u1=exsinexdxu1(x)=cosex

Write the expression to find the arbitrary function u2 ,

u2=G(x)y1y1y2y2y1

Here,

G(x) is the expression for R

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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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