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Math Help - Differential equation problem

  1. #1
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    Differential equation problem

    Hello,

    I'm trying to find the integration factor \lambda(x,y) of this differential equation: (siny-3x^2cosy)cosydx+xdy=0, with exception that \lambda being a function, dependent in only one variable, x or y.

    Any ideas on solving this equation?

    Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Differential equation problem

    Quote Originally Posted by patzer View Post
    Hello,

    I'm trying to find the integration factor \lambda(x,y) of this differential equation: (siny-3x^2cosy)cosydx+xdy=0, with exception that \lambda being a function, dependent in only one variable, x or y.

    Any ideas on solving this equation?

    Thanks
    Hint: If Pdx+Qdy=0 and \lambda(x,y)=\mu (z), the equality (\mu P)_y=(\mu Q)_x sometimes (only sometimes) allows to predict the form of z. Take into account that in general, if we don't know a priori the form of the integration factor, there is no a general method to find it.
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  3. #3
    Junior Member
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    Re: Differential equation problem

    Hello sir,

    Thank you for your help.

    This is my progress so far:

    I know that \frac{d\lambda}{\lambda}=-\frac{\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}}{P}dy where \frac{\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}}{P}= \frac{3x^2sin(2y)+cos(2y)-1}{3x^2(cosy)^2-cosysiny}

    My problem now is with this expression \frac{3x^2sin(2y)+cos(2y)-1}{3x^2(cosy)^2-cosysiny}, I simply don't know how to simplify it.
    Wolfram Alpha gives 2tan(y) as the answer. Can you help me on this point?

    Thank you again.
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