 # Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y ″ − 3 y ′ + 2 y = e x + sin x ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621 ### Calculus (MindTap Course List)

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

#### Solutions

Chapter
Section
Chapter 17.2, Problem 15E
Textbook Problem

## Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y ″ − 3 y ′ + 2 y = e x + sin x

Expert Solution
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

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