Implicit form to explicit form $\frac{e^{2y}}{y}= e^{5x} \Big (\frac{x}{5} + \frac{1}{25}\Big) + C$

Question

I'm trying to write this implicit solution in explicit form. The $y$ in the denominator is messing me up. Sorry about the formatting.

$$\frac{e^{2y}}{y}= e^{5x} \Big (\frac{x}{5} + \frac{1}{25}\Big) + C$$

I need to solve for $y$. Maybe I'm not seeing the algebraic technique. I tried to take the $\ln$ of both sides but then ended up with $\dfrac{2y}{\ln y}$. Any idea how I can isolate the $y$?

Thanks.

Answer 1

Basically, you need to solve for $y$ $$e^{2y}=Ay$$ The only explicit solution is given in terms of Lambert function. $$y=-\frac{1}{2} W\left(-\frac{2}{A}\right)$$ The wikipedia page shows many examples of the manipulations to be done.