Second-Order Nonhomogeneous DE

[ohgeecsea]

The Equation:

x^2 (d^2y/dx^2) + 3x (dy/dx) - 3y = x

The boundary conditions:

y(x=1) = 0
y(x=2) = 1

It's been awhile since I took diffeq, but my research has led me to believe that this is not a Cauchy-Euler Equation since it is not equal to 0, it cannot be separated for separation of variables, it cannot be solved using reduction of order because I was not given one solution, and I cannot solve using Laplace transforms because I was not given initial values, just boundary conditions.

I considered dividing by x^2 or x, but that leaves me either with a constant (still not 0) or a term that has both x and y in it, which I wouldn't know how to solve either.

I just re-imaged my laptop so I do not currently have MATLAB, but if I did I would not be quite sure how to go about it. WolphramAlpha didn't read the equation correctly. I have flipped through a friend's differential equation book and that yielded nothing helpful.

It seems likely that this has a simple solution because this homework was intended as a review of diffeq, so I'm sure a large part of the problem is that I'm rusty and have overlooked something. Help?

 

[fzero]

This equation can easily be solved with the method of undetermined coefficients http://en.wikipedia.org/wiki/Method_of_undetermined_coefficients

Give it a shot and come back with questions if you run into trouble.

 

[gomunkul51]

This is an inhomogeneous Cauchy-Euler ODE !

As always with linear ODEs you can find the homogeneous solution to the ODE and add the particular solution to get the full solution.

Technically you can not use the Undetermined Coefficients method, it can only be used with constant coefficient ODEs.

But you can still guess the particular solution (guess Ax).

The rigorous way to solve it is to use substitution: x = exp(z)

http://en.wikipedia.org/wiki/Cauchy–Euler_equation

 

[Iacha]

If you know the homogeneous solutions, I believe you can get the particular one you need by applying the method of variation of parameters which works even in the non constant coefficient case :)

http://en.wikipedia.org/wiki/Variation_of_parameters