Checking answer vs. mathematica (2nd order equidimensional non homog. ODE)

[iqjump123]

Homework Statement

Obtain general solution:

x^2 y''(x)-2 x y'(x)+2 y(x) = x^2+2

Homework Equations

Using Euler Cauchy method, and using variation of parameters

The Attempt at a Solution

Hey all, I have been struggling with this problem since yesterday in obtaining the particular solution.

First of all, I thought I could use the method of undetermined coefficients, and made the guess as y_p=(Ax²+Bx+C)*x (since the initial guess includes the complimentary solution), but noticed everything just goes to 0.

So I used the Variation of Parameters method. The answer I obtained was:
y(x) = c_2 x^2+c_1 x+x^2 log(x)+1-x^2.

However, when I tried solving this DE in wolfram alpha, it gave me
y(x) = c_2 x^2+c_1 x+x^2 log(x)+1. (no x^2 term)

i tried solving this back and forth and rechecking my answer to see what my problem is, but I can't see what is wrong.

Any insight will be helpful!

Thanks so much, as always :)

iqjump123

 

[vela]

Maybe you typed the problem into Wolfram Alpha incorrectly. Mathematica gives me the same solution you found by hand.

 

[iqjump123]

Maybe you typed the problem into Wolfram Alpha incorrectly. Mathematica gives me the same solution you found by hand.

Thanks for the *lightning fast* reply, vela!
I will go ahead and check it out- but i think the fact that you got the same answer gives me assurance :)

 

[SammyS]

The solutions are the same. The constant c₂ is different in the two solutions.

In the WolframAlpha solution, call it C₂' .

Then let C₂' = c₂ - 1, where c₂ refers to the answer you obtained.

So basically, c₂x² of the WolframAlpha solution has absorbed c₂x² - x² of your solution.

 

[vela]

*facepalm*

 

[SammyS]

*facepalm*

Come on vela. You didn't actually say that the WolframAlpha answer was wrong. You merely confirmed OP's solution.

 

[HallsofIvy]

Since, according to you, the problem specifically says "Using Euler Cauchy method, and using variation of parameters", why do you then say "I thought I could use the method of undetermined coefficients"?

 

[iqjump123]

Since, according to you, the problem specifically says "Using Euler Cauchy method, and using variation of parameters", why do you then say "I thought I could use the method of undetermined coefficients"?

First of all, vela- may I ask why you "facepalm"ed?

Second, HallsofIvy, I mentioned the question as such, because I ended up using variation of parameters, but I thought I could use the method of undetermined coefficients.
After all, using the method of u. c. is an easier alternative if the non homogenous solution is in the standard form. I was merely asking for some insight on if the reason I couldn't use MUC (If I can't)?

thanks.

 

[vela]

I should've noticed the two solutions were actually the same, just written differently.

As far as I know, the method of undetermined coefficients only works when the DE has constant coefficients.

 

[iqjump123]

I thought you were facepalming to my reaction lol. Anyways- good point about the conditions on using m.u.c- it actually seems that v of p is the way that works.

Thanks to all who contributed:)