# 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

### Calculus (MindTap Course List)

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

Chapter
Section

### 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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