Difficult differential equation

[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.

 

[vanhees71]

separation of variables should do.

 

[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.

 

[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).

 

[blue_raver22]

I understand the difficulty of solving complex differential equations. In order to solve this particular equation, we can try using numerical methods such as Euler's method or Runge-Kutta methods. These methods approximate the solution by breaking the equation into smaller steps and calculating the value of y at each step. Another approach could be to use separation of variables, where we separate the equation into two separate equations and then integrate them to find the solution. However, this method may not always be possible for complex equations. Additionally, we could also try using computer software such as MATLAB or Mathematica to solve the equation numerically or symbolically. These software programs have built-in functions and algorithms specifically designed for solving differential equations. Ultimately, the best method for solving this equation will depend on the specific constants and initial conditions, and it may require a combination of approaches.