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Math Help - How to simplify/solve this differential equation

  1. #1
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    How to simplify/solve this differential equation

    I have the following equation

    \frac{\partial}{\partial y}\left(y\frac{dm}{dx}+m\frac{dy}{dx}\right)-\frac{dm}{dx}=0

    where m is a function of y (say m=f\left(y\right)) and y is a function of x (say y=g\left(x\right)). Are there any conditions under which \frac{dm}{dx} becomes identically zero and hence this equation can be reduced to the follwoing form which is easier to solve:

    \frac{\partial}{\partial y}\left(m\frac{dy}{dx}\right)=0

    If such condtions do not exist, what is the best and easiest method to solve the original equation?
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  2. #2
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    Re: How to simplify/solve this differential equation

    If m= f(y) and y= g(x), then \frac{dm}{dx}= \frac{dm}{dy}\frac{dy}{dx} so the answer to your question is "No". That product will be 0 if and only if at least one of \frac{dm}{dy} or \frac{dy}{dx} is 0- in other words if m is NOT a function of y or y is NOT a function of x. Your real problem is that you have two "unkowns", m as a funciton of y and y as a function of x, but only one equation.
    Thanks from JulieK
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  3. #3
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    Re: How to simplify/solve this differential equation

    Many thanks!

    I have two equations not just one because I know f(y) and I want to find g(x) which is the function of interest to me.
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  4. #4
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    Re: How to simplify/solve this differential equation

    I also wish to know if this equation can be solved numerically for g(x) if analytical solution is not possible.
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