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Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621
Chapter 17.2, Problem 18E
Textbook Problem
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Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients.

y + 4 y = e 3 x + x sin 2 x

To determine

To write:

A trial solution for the method of the undetermined coefficients.

Explanation of Solution

1) Concept:

i) General solution of ay''+by'+cy=0:

Roots of ar2+br+c=0 General solution 
r1, r2  are real and distinct y=c1er1x+c2er2x
r1=r2=r y=c1erx+c2xerx
r1 , r2 complex :α±iβ y=eαx(c1cosβx+c2sinβx

ii) undetermined coefficients method:

i) If Gx=ekxPx,  where P  is a polynomial of degree n, then try yPx=ekxQx,  where Qx  is nth-degree polynomial.

ii) If Gx=ekxPxcosmx or Gx=ekxPxsinmx  where P  is a polynomial of degree n, then try yPx=ekxQxcosmx+ekxRxsinmx,  where Qx  and Rx nth-degree polynomial.

iii) If any term of yP  is a solution of the complementary equation, multiply yP  by x  or by x2  if necessary.

2) Given:

y''+4y=e3x+xsin2x 

3) Calculations:

Consider the given differential equation y''+4y=e3x+xsin2x

The auxiliary equation of the homogeneous differential equation is r2+4=0

Solving the auxiliary equation r=+2i

r=2i and r=-2i

Therefore, by concept (i) the complementary solution is yc=C1cos2x+C2sin2x

Let Gx  be e3x+xsin2x

Letting G1x=e3x  has of the form G1x=ekxPx

Where k=3  and Px=1

By using the concept ii),

The trial solution of the form

yPx=ekxQx  where Qx  is the constant polynomial

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Chapter 17 Solutions

Calculus (MindTap Course List)
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Ch. 17.1 - Solve the differential equation. 2d2ydt2+2dydty=0Ch. 17.1 - Solve the differential equation....Ch. 17.1 - Solve the differential equation....Ch. 17.1 - Graph the two basic solutions along with several...Ch. 17.1 - Graph the two basic solutions along with several...Ch. 17.1 - Graph the two basic solutions along with several...Ch. 17.1 - Solve the initial-value problem....Ch. 17.1 - Solve the initial-value problem....Ch. 17.1 - Solve the initial-value problem....Ch. 17.1 - Solve the initial-value problem....Ch. 17.1 - Solve the initial-value problem....Ch. 17.1 - Solve the initial-value problem....Ch. 17.1 - Solve the initial-value problem....Ch. 17.1 - Solve the initial-value problem....Ch. 17.1 - Solve the boundary-value problem, if possible....Ch. 17.1 - Solve the boundary-value problem, if possible....Ch. 17.1 - Solve the boundary-value problem, if possible....Ch. 17.1 - Solve the boundary-value problem, if possible....Ch. 17.1 - Solve the boundary-value problem, if possible....Ch. 17.1 - 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A spring with an 8-kg mass is kept stretched 0.4 m...Ch. 17.3 - A spring with a mass of 2 kg has damping constant...Ch. 17.3 - A force of 13 N is needed to keep a spring with a...Ch. 17.3 - For the spring in Exercise 3, find the mass that...Ch. 17.3 - For the spring in Exercise 4, find the damping...Ch. 17.3 - A spring has a mass of 1 kg and its spring...Ch. 17.3 - A spring has a mass of 1 kg and its damping...Ch. 17.3 - Suppose a spring has mass m and spring constant k...Ch. 17.3 - As in Exercise 9, consider a spring with mass m,...Ch. 17.3 - Show that if 0, but /0 is a rational number, then...Ch. 17.3 - Consider a spring subject to a frictional or...Ch. 17.3 - A series circuit consists of a resistor with R=20,...Ch. 17.3 - A series circuit contains a resistor with R=24 an...Ch. 17.3 - The battery in Exercise 13 is replaced by a...Ch. 17.3 - The battery in Exercise 14 is replaced by a...Ch. 17.3 - Verify that the solution to Equation 1 can be...Ch. 17.3 - The figure shows a pendulum with length L and the...Ch. 17.4 - 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