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Math Help - Series Solution

  1. #1
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    Series Solution

    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)

    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.
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  2. #2
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    Quote Originally Posted by PvtBillPilgrim View Post
    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)

    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.
    Your problem is the sine and cosine functions. Since you are approximating anyway, I'd suggest writing the series forms for the trig functions as you are solving for the coefficients.

    -Dan
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