Particular solution to y″−3y′+2y=2ex

Agohofidov6

Agohofidov6

Answered

2022-01-21

Particular solution to y3y+2y=2ex

Answer & Explanation

limacarp4

limacarp4

Expert

2022-01-21Added 39 answers

y3y+2y=2ex.
Use operator D:
Let D=ddx.
So, (D23D+2)yp=2ex(D1)(D2)yp=2ex
yp=1(D1)(D2)2ex
=2ex1(D+11)(D+12)1=2ex1D(D1)1
yp=2ex1D(01)1=2ex1D1=2xex
ambarakaq8

ambarakaq8

Expert

2022-01-22Added 31 answers

This is too long for a comment, so I posted it as an answer. First solve for the homogeneous equation y3y+2y=0 by setting the right hand side to be zero. The auxiliary equation is m23m+2=0, which has roots m=2,1. Therefore the solution for this homogeneous equation is ex and e2x. Now we want to find a particular solution y3y+2y=2ex. Normally we set the particular solution to be Aex. However, it duplicates with the solution of the homogeneous solution, therefore, we multiple it with x until no duplication occurs. Therefore, the particular solution is given by Axex.
Let me do another example: to solve y2y+y=2ex.
First solve the homogenous equation y2y+y=0. The auxiliary equation is m22m+1=0 which has double roots m=1. Therefore, the solution for this homogeneous equation is ex and xex. Now if we want to find a particular solution y2y+y=2ex. Normally we set the particular solution to be Aex. However, it duplicates with the solution of the homogeneous solution, therefore, we multiple it with x and it becomes Axex, but it still duplicates with xex. Therefore, we mupltiply it by x2, and the particular solution is given by Ax2ex.

RizerMix

RizerMix

Expert

2022-01-27Added 437 answers

The simplest thing in your case is probably to use this (easily justified) trick: If p(x) is a one variable polynomial with complex coefficients and c a complex number, then the ODE p(ddx)y=ecx has a solution of the form ecxq(x) where q(x) is a polynomial whose degree is the number of roots of p(x) equal to c. In you case, p(x)=(x1)(x2),c=1, so the trick tells you to look for a solution of the form (ax+b)ex, and you can clearly assume b=0.

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