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Thread: 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...

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

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

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

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

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

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

    ... and remembering that is...

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

    ... we arrive to write...

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

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

    $\displaystyle a_{0}=1$

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

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

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

    ... so that is...

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

    To be continued in succesive post...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
    Last edited by chisigma; Sep 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' $\displaystyle \varphi_{o} (x)$ we proceed as the $\displaystyle \varphi_{e} (x)$ and obtain...

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

    ... and...

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

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

    $\displaystyle b_{0}=1$

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

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

    ... so that is...

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

    The conclusion is that the DE...

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

    ... has general solution...

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

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

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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