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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 or initial-value problem using the method of undetermined coefficients.

8. y"y' = xe2x, y(0) = 0, y'(0) = 1

To determine

To solve: The differential equation by the method of undetermined coefficients.

Explanation

Given data:

The differential equation is,

yy=xe2x (1)

with y(0)=0,andy(0)=1 .

Consider the auxiliary equation is,

r21=0

r2=1 (2)

Roots of equation (2) are,

r=±1

Write the expression for the complementary solution,

yc(x)=c1ex+c2ex

The particular solution yp(x) is,

yp(x)=(Ax+B)e2x (3)

Differentiate equation (3) with respect to x,

yp(x)=ddx((Ax+B)e2x)

yp(x)=(2Ax+A+2B)e2x (4)

Differentiate equation (3) with respect to x,

yp(x)=ddx((2Ax+A+2B)e2x)

yp(x)=(4Ax+4A+4B)e2x (5)

Substitute equations (3) and (5) in equation (1),

(4Ax+4A+4B)e2x(Ax+B)e2x=xe2x

(3Ax+4A+3B)e2x=xe2x

3Ax+4A+3B=x (6)

Differentiate the equation (6) with respect to x.

ddx(3Ax+4A+3B)=ddx(x)ddx(3Ax)+ddx(4A)+ddx(3B)=ddx(x)3A+0+0=13A=1

Simplify the equation.

A=13

Substitute 0 for x in equation (6),

3A(0)+4A+3B=0

4A+3B=0 (7)

Substitute 90° for x in equation (6),

(4A2B)cos90°+(2A+4B)sin90°=sin90°(4A2B)(0)+(2A+4B)(1)=(1)

2A+4B=1 (8)

Substitute 13 for A in equation (8)

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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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