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Series Solution

  1. May 19, 2007 #1
    1. The problem statement, all variables and given/known data
    Find the indicated coefficients of the power series solution about x = 0 of the differential equation:
    y'' - (sinx)y = cosx, y(0) = -5, y'(0) = 3.
    y = _ + _x + _x^2 + _x^3 + _x^4 + O(x^5)

    2. Relevant equations

    3. The attempt at a solution
    This is going to be a tad confusing in typing it, but I hope it can be read.

    I have the summation(anx^n)
    This equals y.
    y' = summation(nanx^n-1)
    y'' = summation (n(n-1)anx^n-2)

    Just to make it easier, we end up with
    x^n[(n+2)(n+1)an+2-ansinx] = cosx
    Thus, an+2 = (cosx + ansinx) / ((n+2)(n+1))

    I know obviously that the first two terms (x^0 and x^1) are -5 and 3 respectively. I also know that the x^2 term is 0.5 by plugging in 0 for x. However, this doesn't work for the rest of them. I've done a lot of these types of problems, but this is the first one with sin(x) or cos(x), which puts an "x" in the an+2 equation (which I wrote above). What does x equal in this case? Can anyone just show me how to find the remaining coefficients because I'm pretty sure my equation is correct.
  2. jcsd
  3. May 19, 2007 #2


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    Science Advisor

    Expand sin(x) and cos(x) in Taylor's series.
  4. May 19, 2007 #3
    Yes, but I still don't have x do I?
  5. May 19, 2007 #4
    I got it. Thanks for the help.
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