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Series Solution for O.D.E (Frobenius method?)

  1. Feb 16, 2013 #1
    1. The problem statement, all variables and given/known data
    (For Physics 306: Theoretical methods of Physics... Text book: Mathematical Tools for Physics (really good!)...
    Assumption: 300 level class = considerably junior level class)

    -- Find a series solution about x=0 for y''+ysec(x) = 0, at least to a few terms.

    (Ans: a[itex]_{0}[/itex][1-(1/2)x[itex]^{2}[/itex]+0x[itex]^{4}[/itex]+1/720x[itex]^{6}[/itex]+...]+a[itex]_{1}[/itex][x-(1/6)x[itex]^{3}[/itex]-(1/60)x[itex]^{5}[/itex]+...]


    2. Relevant equations

    Frobenius Solution: Supposing there is a singularity point at x=x0, a possible solution is given by the power series, that is

    Ʃ(from k=0 to inf) a[itex]_{k}[/itex]x[itex]^{k+s}[/itex]


    3. The attempt at a solution

    There is a singularity point not at x=0, but at x= npi/2. The problem specifies, though, to try finding a power series solution about x=0.

    Let x(t) = Ʃ a[itex]_{k}[/itex]x[itex]^{k+s}[/itex]

    Using this, plug it into the equation to obtain

    Ʃ a[itex]_{k}[/itex]x[itex]^{k+s-2}[/itex] (k+s)(k+s-1) +Ʃ a[itex]_{k}[/itex]x[itex]^{k+s}[/itex] = 0

    after that I matched the terms up by pushing the first summation up by two indicese. To do this, I wrote out two terms, then shifted the k values by +2.

    a[itex]_{0}[/itex](s)(s-1)x[itex]^{s-2}[/itex]+a[itex]_{1}[/itex](1+s)(s)x[itex]^{s-1}[/itex]+Ʃ a[itex]_{k+2}[/itex]x[itex]^{k+s}[/itex] (k+s+2)(k+s+1) +Ʃ a[itex]_{k}[/itex]x[itex]^{k+s}[/itex] = 0

    This is where I stopped. I'm unsure as to what to do next. You match up the terms in a regular Forbenius method, but this time it has a cos(x) in the second term, which is what makes this problem strange to me.

    I think the next step is to find s, then just write out a few terms of both the series.

    Also sorry about the mess regarding the equations. I got rid of all the sigma indices, so when you see a sigma, it is always from k=0 to k=∞.
     
  2. jcsd
  3. Feb 17, 2013 #2
    Expand the secant function into a power series.
     
  4. Feb 17, 2013 #3

    vela

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    Since there's no singularity at x=0, you can use a plain old power series and avoid trying to find s from an indicial equation.
     
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