Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis
Differential Equations: Computing and Modeling (5th Edition), Edwards, Penney & Calvis
5th Edition
ISBN: 9780321816252
Author: C. Henry Edwards, David E. Penney, David Calvis
Publisher: PEARSON
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Chapter 5.7, Problem 1P
Program Plan Intro

Program Description: Purpose of problem is to find a particular solution of the system x=x+2y+3 and y=2x+y2 by using method of undetermined coefficient.

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Explanation of Solution

Given information:

The system of differential equation is x=x+2y+3 and y=2x+y2 .

Explanation:

Consider matrix form of the system of differential equation as,

  x=[1221]x+[32] ...............(1)

Consider [1221] as matrix A and non homogeneous term f=[32]

The non homogeneous term is a constant vector.

Therefore, A constant particular solution is in the form as,

  xp(t)=[ a 1 a 2]

Substitute [ a 1 a 2] for x and [00] for x in equation (1)

  [00]=[ 1 2 2 1][ a 1 a 2]+[ 3 2][00]=[ a 1+2 a 22 a 1+ a 2]+[ 3 2][00]=[ a 1+2 a 2+32 a 1+ a 22]

Equate the coefficient to obtain first equation.

  a1+2a2+3=0a1=32a2

The second equation is obtained as,

  2a1+a22=0

Substitute 32a2 for a1 in above equation.

  2(32a2)+a22=064a2+a22=03a2=8a2=83

The value of a1 is obtained as,

  a1=32(83)=3+163=9+163=73

The particular solution of the system of differential equation is,

  xp=[x(t)y(t)] ...............(2)

Substitute 73 for x(t) and 83 for y(t) in equation (2)

  xp=[ 7 3 8 3]

Conclusion:

Thus, the particular solution of the differential system x=x+2y+3 and y=2x+y2 is x(t)=73 and  y(t)=83 .

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