How to solve this differential equation
$\displaystyle \left(\frac{dy}{dx}\right)^{4}+B\left(\frac{dy}{dx }\right)^{3}+C=0$
where $\displaystyle B$ and $\displaystyle C$ are constants.
let dy/dx = u
solve that nasty 4th order polynomial equation you now have, for u. (it's not pretty)
Note that u is just a constant. (up to 4 constants actually)
Integrate u over x to obtain y(x) = u x + const. This is actually up to 4 separate solutions as there are up to 4 distinct roots of your polynomial.
The differential equation is satisfied for all the u's.
did you maybe mean d^{4}y/dx^{4} + B d^{3}y/dx^{3} + C = 0 ?