Results 1 to 5 of 5

Math Help - n-th order ODE, general solution?

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    18

    n-th order ODE, general solution?

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

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

    where 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
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,364
    Thanks
    39
    Quote Originally Posted by DJDorianGray View Post
    Hi everyone,
    I need to find the general solution of the following:

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

    where 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 k. What I would suggest is to look at n = 1, 2, 3 etc, and solve and then look for patterns.

    Yes -a very nice pattern!
    Last edited by Jester; February 10th 2010 at 02:25 PM. Reason: update
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2009
    Posts
    18
    I can solve for n = 1, but n = 2 is already quite complicated. And even if I had a pattern, how do I generalize?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,364
    Thanks
    39
    Quote Originally Posted by DJDorianGray View Post
    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

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

     <br />
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}<br />

    Do you see a pattern?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Aug 2008
    Posts
    903
    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.

    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:

    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.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. General solution to 2nd order
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: October 8th 2010, 10:36 AM
  2. General Solution to Second Order Differntial Eq
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: May 8th 2010, 02:47 PM
  3. General Solution 2nd order
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 5th 2010, 03:26 PM
  4. the general solution to a first order ODE
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: March 12th 2010, 07:08 AM
  5. 2nd order PDE- general solution
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: November 9th 2009, 08:02 AM

Search Tags


/mathhelpforum @mathhelpforum