How to simplify/solve this differential equation

I have the following equation

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

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

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

If such condtions do not exist, what is the best and easiest method to solve the original equation?

Re: How to simplify/solve this differential equation

If m= f(y) and y= g(x), then $\displaystyle \frac{dm}{dx}= \frac{dm}{dy}\frac{dy}{dx}$ so the answer to your question is "No". That product will be 0 if and only if at least one of $\displaystyle \frac{dm}{dy}$ or $\displaystyle \frac{dy}{dx}$ 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.

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.

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.