# Thread: n-th order ODE, general solution?

1. ## n-th order ODE, general solution?

Hi everyone,
I need to find the general solution of the following:

$\displaystyle x^{2n} \left( \frac{d}{dx} - \frac{a}{x} \right)^n y = ky$

where $\displaystyle n$ is a positive integer. Any help would be appreciated, I don't know where to start from (finding out where to start, at least in this problem, seems like the hard part). Thanks

2. Originally Posted by DJDorianGray
Hi everyone,
I need to find the general solution of the following:

$\displaystyle x^{2n} \left( \frac{d}{dx} - \frac{a}{x} \right)^n y = ky$

where $\displaystyle n$ is a positive integer. Any help would be appreciated, I don't know where to start from (finding out where to start, at least in this problem, seems like the hard part). Thanks
First, tell us about $\displaystyle k$. What I would suggest is to look at $\displaystyle n = 1, 2, 3$ etc, and solve and then look for patterns.

Yes -a very nice pattern!

3. I can solve for n = 1, but n = 2 is already quite complicated. And even if I had a pattern, how do I generalize?

4. Originally Posted by DJDorianGray
I can solve for n = 1, but n = 2 is already quite complicated. And even if I had a pattern, how do I generalize?
This is what I got for n = 1, 2 and 4

$\displaystyle n=1,\;\;\;y = c_1 x^a e^{-k/x}$
$\displaystyle n = 2,\;\;\;y = c_1 x^{a+1} \sinh \frac{\sqrt{k}}{x} + c_2 x^{a+1} \cosh \frac{\sqrt{k}}{x}$

$\displaystyle n = 4,\;\;\;y = c_1 x^{a+3} \sinh \frac{k^{1/4} }{x} + c_2 x^{a+3} \cosh \frac{k^{1/4} }{x} + c_3 x^{a+3} \sinh \frac{i k^{1/4} }{x} + c_4 x^{a+3} \cosh \frac{i k^{1/4} }{x}$

Do you see a pattern?

5. If it get's too cumbersome, you can if you wish work on in in Mathematica using this code:

Code:
nval = 1;
myop = D[#1, x] - (a/x)*#1 &
myeqn = FullSimplify[x^(2*nval)*
Nest[myop, f[x], nval]] == k*f[x]
FullSimplify[DSolve[myeqn, f, x]]
Here's n=3 using the code although I haven't checked it throughly and where I've left the Mathematica-specific notation in since I just copied it directly over so I didn't have to type in the latex.

$\displaystyle x^{2+a} \left(e^{-\frac{k^{1/3}}{x}} C[1]+e^{\frac{(-1)^{1/3} k^{1/3}}{x}} C[2]+e^{-\frac{(-1)^{2/3} k^{1/3}}{x}} C[3]\right)$

As a check of the code, here is n=2 which I believe is the same as Danny's solution:

$\displaystyle e^{\frac{\sqrt{k}}{x}} x^{1+a} C[1]+\frac{e^{-\frac{\sqrt{k}}{x}} x^{1+a} C[2]}{2 \sqrt{k}}$

Double-check everything though ok.