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Differential Equation (cosx)y +y=0 using power/taylor series

  1. Jul 22, 2011 #1
    Differential Equation (cosx)y"+y=0 using power/taylor series

    1. The problem statement, all variables and given/known data
    Find the first three nonzero terms in each of two linearly indepdent solutions of the form y=[itex]\sum(c_{n}x^{n})[/itex]. Substitute known Taylor series for the analytic functions and retain enough terms to compute the necessary coefficients.


    (cosx)y"+y=0

    2. Relevant equations



    3. The attempt at a solution

    I substituted the taylor series for cosx, and the general power series into the equation, to get...

    [itex]\sum(\frac{(-1)^{n}x^{2n}}{(2n)!})[/itex][itex]\sum(n(n-1)c_{n}x^{n-2})[/itex]+[itex]\sum(c_{n}x^{n})[/itex]=0

    I then multiplied the first term out by series multiplication but that did not seem to get me anywhere. I am stuck at this part

    By the way... for the second series (second derivative) the index begins at n=2 and the other two are n=0....sorry i do not know how to put the subscripts on the series
     
  2. jcsd
  3. Jul 23, 2011 #2

    ehild

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    Re: Differential Equation (cosx)y"+y=0 using power/taylor series

    Use different summation indices in the different sums and collect the terms containing xn from the product.

    ehild
     
  4. Jul 23, 2011 #3
    Re: Differential Equation (cosx)y"+y=0 using power/taylor series

    LaTeX or Math Symbol Notation
    On my web browser, there's a sigma/sum/ [itex]\sum[/itex] sign at the right end of the row with, Bold, Italic, underline. Click it to open a whole slew of notations. This box lists [itex]\sum[/itex] on the operator submenu. I personally, use the Subscript and Superscript menu options. One can also right click on someone's notation to "show source" and see how individual code works. Somewhere, there's a thread for the sole purpose of explaining the notation. Finally, LaTeX is standardized, so in theory it is possible to download a LaTeX editor to create the markup code, then copy and paste into here.

    The lack of clarity for index variable, initial value, and final value is throwing me.

    [itex]\sum ^{i=0}_{n} or
    \sum ^{n=1}_{m}[/itex]


    And yes, IIRC, ehild has the right idea. For the 2nd derivative series, substitute,
    [itex]n_{a}=n_{b}\: -2[/itex]
    so that all sums have the same initial value.

    However, I don't remember, is it ordered on highest degree first or lowest degree first?
    For the cos x series,
    x^(2n), x^(2n-2), or x^0, x^2, x^4?
     
  5. Jul 23, 2011 #4
    Re: Differential Equation (cosx)y"+y=0 using power/taylor series

    So if i change the summation index for the second derivative i get....

    [itex]\sum^{\infty}_{n=0}(n+2)(n+1)c_{n+2}x^{n}[/itex]

    and the first power series reamins the same. Also for cosx it is


    [itex]\sum^{\infty}_{n=0}\frac{(-1)^{n}(x)^{2n}}{(2n)!}[/itex]



    However, once i get to this stage... do i multiply out the cosx and y"? Or should i simply just take n=0,n=1,n=2...etc. of the cosx and y" and multply those out and compare coefficients to solve for the c's?
     
    Last edited: Jul 23, 2011
  6. Jul 24, 2011 #5
    Re: Differential Equation (cosx)y"+y=0 using power/taylor series

    I figured it out. Thank you for your help
     
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