Variation of Parameters - Higher order DE

[Wellesley]

Homework Statement

Given that x, x² and 1/x are solutions of the homogeneous equation corresponding to:


<br /> x^3y&#039;&#039;&#039; + x^2y&#039;&#039;-2xy&#039;+2y=2x^4 <br />

x>0
determine a particular solution.

Homework Equations

The Attempt at a Solution

I'm trying to solve this problem using three simultaneous equations:

<br /> u_1&#039;x + u_2&#039;x^2 + u_3&#039;*(1/x) = 0<br />

u_1&#039; + 2u_2&#039;x - u_3&#039;x^{-2}= 0

2u_2&#039; + 2u_3&#039;x^{-3}= 2x^4

Then, when I try and solve for u₁, u₂ and u₃, I seem to doing it incorrectly, the wrong answer. I've come back to this problem several times over the last three days, and all that has changed were the answers I'm getting! Should I be trying a different method in solving this problem?

Thanks.

 

[Dick]

Variation of parameters is the hard way. The easy way is to try and guess a solution. I'm guessing that there is a solution of the form a*x^4. Do you see why?

 

[Wellesley]

Thanks for the response.

I do see what you are saying about the guess, and it works. ax⁴ comes from 2x⁴. Plugging in the derivatives of ax⁴ into the original equation, you can solve for a...a=1/15 so the answer is x⁴/15 which is the right answer.

Will this process be okay for a homework problem in the variation of parameters chapter? I just don't want to get marked down for not using the 'correct' method. When would you use the variation of parameters?

 

[Dick]

Thanks for the response.

I do see what you are saying about the guess, and it works. ax⁴ comes from 2x⁴. Plugging in the derivatives of ax⁴ into the original equation, you can solve for a. a=1/15 so the answer is x⁴/15 which is the right answer.

Will this process be okay for a homework problem in the variation of parameters chapter? I just don't want to get marked down for not using the 'correct' method.

Hard to say. But if they say 'use variation of parameters' then I guess you had better use variation of parameters.

 

[Wellesley]

Hard to say. But if they say 'use variation of parameters' then I guess you had better use variation of parameters.

Well, the problem just says to determine a particular solution. They didn't say which method to use, like they did on other problems so I think I'll be fine. Thanks for the help!

 

[Dick]

BTW the your variation of parameters was going wrong because in the technique the coefficient of the highest derivative should be 1. You should have divided both sides of the ODE by x^3 before you started. So the 2*x^4 in your last equation should be 2x. Then it will work.

 

[Wellesley]

BTW the your variation of parameters was going wrong because in the technique the coefficient of the highest derivative should be 1. You should have divided both sides of the ODE by x^3 before you started. So the 2*x^4 in your last equation should be 2x. Then it will work.

Thanks! I knew something was going wrong but I didn't know where.