# Difficult differential equation

1. Nov 18, 2013

### JulieK

I have the following differential equation which I want to solve for $y$ as a function of $x$

$\frac{dy}{dx}=\frac{C_{1}\left(C_{5}y+C_{6}\right)^{2}}{C_{2}\left(C_{3}y+C_{4}\right)-C_{7}\left(C_{5}y+C_{6}\right)^{6}}$

where $C_{1},C_{2},C_{3},C_{4},C_{5},C_{6},C_{7}$ are constants.
Can anyone suggest a method for solving this equation.

2. Nov 18, 2013

### vanhees71

separation of variables should do.

3. Nov 19, 2013

### JulieK

I want y as an explicit function of x not the other way round.
The separation and integration will produce x as a function of y which is not very useful for my purpose.

4. Nov 19, 2013

### PSarkar

You will get $x$ as a function of $y$, say $f(y) = x$. Then you can try and find the inverse to get $y$ as a function of $x$, i.e. $y = f^{-1}(x)$. There are a few things that can go wrong. If $f$ is not invertible then that tells you that there is no unique solution $y(x)$ to the differential equation. Otherwise, another thing that can go wrong is the inverse cannot be written down in terms of elementary functions.

So separation of variables is still the right (only?) approach. It won't change any of the facts above ($f$ invertible and inverse can be written down).