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Few questions about series solution of ODEs

  1. Nov 29, 2013 #1
    Consider the ODE [itex]x(x-1)y''-xy'+y=0[/itex].

    I need help in identifying the method of solution (power series or frobenius) for this ODE.

    Using the formulae [itex]\stackrel{limit}{_{x→x_{o}}}\frac{q(x)+r(x)}{p(x)}[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}\frac{(x-x_{o})q(x)+(x-x_{o})^{2}r(x)}{p(x)}[/itex] , where p(x)=x(x-1), q(x)=-x, and r(x)=1, I have worked out the following:

    My question is: How do I use this data to find out which solution method to use?

    I am guessing that since the problem equation has a regular singular point besides having a singular point, we will drop the power series method and use the Frobenius method of solution. Am I right?

    Another question: Which regular singular point do we choose if there are more than one regular singular points for an ODE?
    Last edited: Nov 29, 2013
  2. jcsd
  3. Nov 30, 2013 #2
    Hi !
    An obvious solution of the ODE is y=x.
    So, in order to find a second solution, let y=x*f(x) where f(x) is an unknown function. Bring it back into the ODE
    This leads to a first order ODE easy to solve, which gives f '(x). The integration leads to f(x)=ln(x)+1/x, then y(x)= 1+x*ln(x)
    The general solution of the ODE is y=a*x+b*(1+x*ln(x)) where a and b are constant.
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