Results 1 to 5 of 5

Math Help - Power Series/Differential Equations

  1. #1
    Junior Member
    Joined
    Sep 2008
    Posts
    58

    Power Series/Differential Equations

    Find the first four nonzero terms in each of two linearly independent power series solutions about the origin.

    What do you expect the radius of convergence to be for each solution?

     (cosx)y^{''} +xy^{'} -2y=0

    Set the coefficients near zero and first terms to 1



    I am not exactly sure what the bolded sentence means. Also, how should I go about starting the problem? Is it best to start writing out the series for each term? like

    Cosx =  1-\frac{x^2}{2!} + \frac{x^4}{4!} + ...

    and so on.

    Thanks for any help!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by Oblivionwarrior View Post
    Find the first four nonzero terms in each of two linearly independent power series solutions about the origin.

    What do you expect the radius of convergence to be for each solution?

     (cosx)y^{''} +xy^{'} -2y=0

    Set the coefficients near zero and first terms to 1



    I am not exactly sure what the bolded sentence means. Also, how should I go about starting the problem? Is it best to start writing out the series for each term? like

    Cosx =  1-\frac{x^2}{2!} + \frac{x^4}{4!} + ...

    and so on.

    Thanks for any help!
    Start by assuming that the solution to this problem is y=\sum_{n=1}^{\infty}a_nx^n. Then substitute, differentiate, adjust indicies so they are all the same, and then solve the ensuing recurrence relation.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Sep 2008
    Posts
    58
    Ok, ill try that out. What about that bolded sentence? Is that just talking about where I begin once I got the recurrence formula?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by Oblivionwarrior View Post
    Ok, ill try that out. What about that bolded sentence? Is that just talking about where I begin once I got the recurrence formula?
    I'm not entirely sure.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Aug 2008
    Posts
    903
    I think the equation \cos(x)y''+xy'-2y=0 would be quite a challenge to solve by substituting \sum_{n=0}^{\infty}a_nx^n because of the \cos(x) term: that's going to involve a Cauchy product of sums for the first term which can be done but is messy. I think in this particular case, just need to differentiate the differential equation to find the first few coefficients: Let y(0)=a and y'(0)=b and write it as:

    y''=-\frac{x}{\cos(x)}y'+\frac{2}{\cos(x)}y

    Now we seek a solution about the origin in which everything is well-behaved until we reach the first singular point at x=\pi/2. Therefore we can expect a solution of the form:

    y(x)=\sum_{n=0}^{\infty}y^{(n)}(0)\frac{x^n}{n!};\  quad R=\pi/2

    We already know what the first two terms are. Now differentiate to get the third term:

    \frac{d}{dx}y''\Bigg|_{x=0}=\frac{d}{dx}\left(-\frac{x}{\cos(x)}y'+\frac{2}{\cos(x)}y\right)_{x=0  }

    and substitute the values for y(0),y'(0),y''(0).

    Do this again to get the fourth term. However that's only one series but I think the two linearly independent solutions are made up of the separate sums of the even and odd powers of x which would mean you'd need to calculate about 8 derivatives. Might get messy and maybe there is another way.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Power series - Differential equation
    Posted in the Differential Equations Forum
    Replies: 4
    Last Post: March 26th 2011, 04:43 AM
  2. Finding differential equation from power series
    Posted in the Differential Equations Forum
    Replies: 1
    Last Post: February 16th 2010, 11:11 PM
  3. Writing a differential equation in power series
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: December 9th 2009, 01:48 AM
  4. Power series and differential equation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 16th 2007, 12:00 PM
  5. Differential equation and power series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 11th 2007, 03:23 PM

Search Tags


/mathhelpforum @mathhelpforum