How to integrate this partial differential equation

JulieK

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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kevinferreira

Yes, you should have
$$m(y)\frac{dy}{dx}=C(x)$$
And therefore you can solve it by
$$m(y)dy=C(x)dx$$
Which you can integrate.

haruspex

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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.
You'll have to use boundary conditions. There's nothing in the equation that gives a clue about the form of C(x).

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