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Math Help - help with y''+ycoshx=0

  1. #1
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    help with y''+ycoshx=0

    i have twisted my mind but don't know!! It's not x=e^t, or y=e^t, and can't think of an appropriate variation of the type y=v(x)f(x).

    what to do?? any hints?
    pepsigirl
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  2. #2
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    Whoa. WolframAlpha gave this intimidating answer. The Mathieu cosine and sine functions appear to be the solutions to this DE. Not very pretty, I'm afraid. And I don't know much more than what I've just written. What class is this for, anyway?
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  3. #3
    MHF Contributor chisigma's Avatar
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    The DE is linear and 'incomplete' and its general solution is...

    y(x)= c_{1}\ \varphi_{e} (x) + c_2 \ \varphi_{o} (x) (1)

    ... where \varphi_{e} (x) and \varphi_{o} (x) are respectively an 'even' and an 'odd' function that we suppose analytic so that is...

    \displaystyle \varphi_{e} (x) = \sum_{n=0}^{\infty} a_{n}\ x^{2n}

    \displaystyle \varphi_{o} (x) = x\ \sum_{n=0}^{\infty} b_{n}\ x^{2n} (2)

    Let's consider first \varphi_{e} (x), for which we can hypothesize that is a_{0}=1. Is...

    \displaystyle \varphi^{''}_{e} (x) = \sum_{n=1}^{\infty} 2n\ (2n-1)\ a_{n}\ x^{2n-2} (3)

    ... and remembering that is...

    \displaystyle \cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} (4)

    ... we arrive to write...

    \displaystyle \varphi_{e} (x)\ \cosh x= \sum_{n=0}^{\infty} x^{2n}\ \sum_{k=0}^{n} \frac{a_{k}}{\{2\ (n-k)\}!} (5)

    Combining (3), (5) and the DE we are able to find the a_{n}...

    a_{0}=1

    2\ a_{1} + a_{0} = 0 \rightarrow a_{1} = - \frac{1}{2}

    \displaystyle 12\ a_{2} + \frac{a_{0}}{2} + a_{1}=0 \rightarrow a_{2} = 0

    \displaystyle 30\ a_{3} + \frac{a_{0}}{6!} + \frac{a_{1}}{4!} + \frac{a_{2}}{2} + a_{3} = 0 \rightarrow a_{3}= \frac{1}{1860}

    ... so that is...

    \displaystyle \varphi_{e} (x)= 1 - \frac{x^{2}}{2} + \frac{x^{6}}{1860} + ... (6)

    To be continued in succesive post...

    Kind regards

    \chi \sigma
    Last edited by chisigma; September 2nd 2010 at 03:44 PM. Reason: Only partial solution found...
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  4. #4
    MHF Contributor chisigma's Avatar
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    Regarding the 'odd function' \varphi_{o} (x) we proceed as the \varphi_{e} (x) and obtain...

    \displaystyle \varphi^{''}_{o} (x) = \sum_{n=1}^{\infty} 2n\ (2n+1)\  b_{n}\ x^{2n-1} (1)

    ... and...

    \displaystyle  \varphi_{o} (x)\ \cosh x = \sum_{n=0}^{\infty} x^{2n+1}\ \sum_{k=0}^{n} \frac{b_{k}}{\{2\ (n-k)\}!} (2)

    Now from (1), (2) and the DE we obtain the b_{n} as follows...

    b_{0}=1

    \displaystyle  6\ b_{1} + \frac{b_{0}}{2} + b_{1} = 0 \rightarrow b_{1} = -\frac{1}{14}

    \displaystyle  20\ b_{2} + \frac{b_{0}}{4!} + \frac{b_{1}}{2} + b_{2} =0 \rightarrow b_{2}= -\frac{1}{3528}

    ... so that is...

    \displaystyle \varphi_{o} (x) = x - \frac{x^{3}}{14} - \frac{x^{5}}{3528} + ... (3)

    The conclusion is that the DE...

    \displaystyle y^{''} + y\ \cosh x =0 (4)

    ... has general solution...

    \displaystyle y(x) = c_{1}\ \varphi_{e} (x) + c_{2}\ \varphi_{o} (x) (5)

    ... where \varphi_{e} (x) and \varphi_{o} (x) have been found in the previous and in the actual post...

    Kind regards

    \chi \sigma
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