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Math Help - How to integrate this partial differential equation

  1. #1
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    How to integrate this partial differential equation

    I have the following equation

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

    where y is a function of x and m is a function of y. If I integrate this equation first with respect to y should I get a function of x as the constant of integration (say C\left(x\right)) or it is just a constant? If it is a function, how can I then find its form (e.g. polynomial, etc.)? Should I use boundary conditions or I can decide about the form from inspecting the type of the equation.
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  2. #2
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    Re: How to integrate this partial differential equation

    Quote Originally Posted by JulieK View Post
    I have the following equation

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

    where y is a function of x and m is a function of y. If I integrate this equation first with respect to y should I get a function of x as the constant of integration (say C\left(x\right)) or it is just a constant? If it is a function, how can I then find its form (e.g. polynomial, etc.)? Should I use boundary conditions or I can decide about the form from inspecting the type of the equation.
    \displaystyle \begin{align*} \frac{\partial }{\partial y} \left( m\, \frac{dy}{dx} \right) &= 0 \\ m\,\frac{dy}{dx} &= \int{0\, dy} \\ m\,\frac{dy}{dx} &= f(x) \\ \frac{dy}{dx} &= \frac{f(x)}{m} \\ y &= \int{\frac{f(x)}{m}\,dx} \\ y &= \frac{1}{m} \int{f(x)\,dx} + C \end{align*}

    Unless you have some boundary conditions now, this is the simplest it's going to get.
    Thanks from JulieK
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