Help solving a nonlinear first order ODE

Question

Is it possible to find an exact solution for the following ODE? $$ \frac{1}{\sqrt{1+y'^2}} + \frac{Ay}{2}-\frac{B}{y}=0$$ Given that $A$ and $B$ are positive constants. Any help is appreciated.

Answer 1

I do not think that you could obtain $y(x)$ but swithching variables, you easuation is $$\frac{1}{\sqrt{1+\frac{1}{x'^2}}} + \frac{Ay}{2}-\frac{B}{y}=0$$ leading to $$x'= \pm \sqrt{\frac{\left(A y^2-2 B\right)^2}{\left(A y^2-2 B\right)^2-4 y^2}}=\pm \frac{A y^2-2 B } {\sqrt{(A y^2-2y-2 B)(A y^2+2y-2 B) } }$$ which is separable.

The result would involve elliptic integrals of the first an second kinds.