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Math Help - Is this sine equation solvable

  1. #1
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    Is this sine equation solvable

    Hi,

    I need to either solve or prove that the resolution of the following ODE is impossible;

    y''(x)=a*y(x)+b*y'(x)+c*sin(d*x)*y(x)

    Mathemtatica refuse solving it rather giving

    " InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses."

    Any help would be highly apreciated, thank you

    Isatis55
    Last edited by isatis55; October 27th 2012 at 12:00 AM.
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  2. #2
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    Re: Is this sine equation solvable

    Hey isatis55.

    I haven't taken DE's for a while but the output suggests that f(x) is not a unique function (with the multi-valued comment) and this implies that y(x) doesn't exist since it has to be unique by definition to give a function (functions must be unique and give one output for every input).
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  3. #3
    Behold, the power of SARDINES!
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    Re: Is this sine equation solvable

    Quote Originally Posted by isatis55 View Post
    Hi,

    I need to either solve or prove that the resolution of the following ODE is impossible;

    y''(x)=a*y(x)+b*y'(x)+c*sin(d*x)*y(x)

    Mathemtatica refuse solving it rather giving

    " InverseFunction::ifun: Inverse functions are being used. Values may be lost for multivalued inverses."

    Any help would be highly apreciated, thank you

    Isatis55
    If you multiply the equation by -e^{-bx}

    This gives

    -e^{-bx} y''+be^{-bx}y'+ae^{-bx}y= -ce^{-bx}\sin(dx) y

    Now the differential equation can be put in its self-adjoint form

    -\frac{d}{dx}\left[ e^{-bx}\frac{dy}{dx}\right]+ae^{-bx}y= -ce^{-bx}\sin(dx) y

    This is a Sturm-Liouville differential equation.

    The boundary or initial conditions determine if the equation has solutions.

    For example if you require y(0)=y'(0)=0

    The equation will have only the trivial solution y=0
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  4. #4
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    Re: Is this sine equation solvable

    Great, thanks a lot.

    I am now altering the initial problem and I am again faced against a non linear differential equation given by

    [math] y''(x)+a*y'(x)^2+b*y'(x)+c*y(x) + c*sin(d*t) + e = 0 [\math]

    Do you think it is solvable?

    PS: I definitely need a course on ODE.
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