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.

 

[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]

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).