Results 1 to 2 of 2

Math Help - Series Solutions Near an Ordinary Point help please

  1. #1
    Newbie
    Joined
    May 2009
    Posts
    7

    Series Solutions Near an Ordinary Point help please

    Hi, I am having trouble solving this problem, please help me figure this one out. Thanks in advance.

    xy'' + y' + xy=0 at ordinary point x=1.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    With the substitution x-1=\xi the equation becomes...

    (1+\xi)\cdot y^{''} + y^{'} + (1+\xi)\cdot y=0 (1)

    That is an 'incomplete' linear ODE and we will search an analytic solution written as...

    y(\xi)= \sum_{n=0}^{\infty} a_{n}\cdot \xi^{n} (2)

    If we derive from (2) le derivatives of y(\xi) and subsitute them in (1) we arrive to write the following 'infinite sysytem' of algebric equations...

    a_{n-3} + a_{n-2} + n\cdot (n-1)\cdot (a_{n-1} + a_{n})=0 (3)

    ... whose solution is...

    a_{n}= -a_{n-1} - \frac{ a_{n-2} + a_{n-3}}{n\cdot (n-1)} (4)

    Since a solution of (1) multiplied by a constant is also a solution of (1), we can set without limitations a_{0}=1 and so with (4) we derive...

    a_{1}= -1, a_{2}= \frac{1}{2}, a_{3}= -\frac{1}{2}, a_{4}= \frac{13}{24}, a_{5}= - \frac{13}{24}, ...

    The searched solution od (1) is then...

    y(x)= 1 - (x-1) + \frac{1}{2}\cdot (x-1)^{2} - \frac{1}{2}\cdot (x-1)^{3} + \frac{13}{24}\cdot (x-1)^{4} - \frac{13}{24}\cdot (x-1)^{5} + ... (5)

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: February 9th 2010, 04:15 PM
  2. Ordinary 2nd Ordinary Non-homogenous DE
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 16th 2009, 08:24 PM
  3. Series Solutions Near a Regular Singular Point
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 20th 2009, 05:48 AM
  4. Ordinary point for ODE
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 23rd 2009, 07:19 AM
  5. Replies: 2
    Last Post: January 8th 2008, 03:33 PM

Search Tags


/mathhelpforum @mathhelpforum