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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.

6. y" – 4y' + 4y = x – sin x

To determine

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

Explanation

Given data:

The differential equation is,

y4y+4y=xsinx (1)

Consider the auxiliary equation is,

r24r+4=0 (2)

Roots of equation (2) are,

r=4±(4)24(1)(4)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=4±02=2

Write the expression for the complementary solution,

yc(x)=c1e2x+c2xe2x

Re-write equation (1) by neglecting the term sinx ,

y4y+4y=x (3)

The particular solution yp1(x) is,

yp1(x)=Ax+B (4)

Differentiate equation (4) with respect to x,

yp1(x)=ddx(Ax+B)

yp1(x)=A (5)

Differentiate equation (5) with respect to x,

yp1(x)=ddx(A)

yp1(x)=0 (6)

Substitute equations (4), (5) and (6) in (3),

04A+4(Ax+B)=x

4Ax+(4B4A)=x (7)

Differentiate the equation (7) with respect to x,

ddx(4Ax+4B4A)=ddx(x)ddx(4Ax)+ddx(4B)+ddx(4A)=ddx(x)4A+0+0=14A=1

Simplify the equation,

A=14

Substitute 0 for x in equation (7),

4A(0)+(4B4A)=0

4B4A=0 (8)

Substitute 14 for A in equation (8),

4B4(14)=04B1=04B=1B=14

Substitute 14 for A and 14 for B in equation (4),

yp1(x)=14x+14 (9)

Re-write equation (1) by neglecting x.

y4y+4y=sinx (10)

The particular solution yp2(x) is,

yp2(x)=Acosx+Bsinx (11)

Differentiate equation (11) with respect to x,

yp2(x)=ddx(Acosx+Bsinx)

yp2(x)=Asinx+Bcosx (12)

Differentiate equation (12) with respect to x,

yp2(x)=ddx(Asinx+Bcosx)

yp2(x

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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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