Implicit form of general equation

Question

Find, in implicit form, the general solution of the differential equation: $$\frac{dy}{dx}= \frac{2y^4e^{2x}}{3\left(e^{2x}+7\right)^2}$$

I am struggling to make any sense of this. What I have understood is that first I need to seperate the variables then integrate but I am not sure how to seperate the variables.

The equations I have are : dy/dx=f(x)g(y) then divide both sides by g(y) to get: 1/g(y) dy/dx=f(x)

I am just not sure which part of the equations would be the g(y) and f(x) pary. Any help greatly appreciated!

Answer 1

Let $y=y(x)$, then $$ \frac{dy}{dx}= \frac{2y^4e^{2x}}{3\left(e^{2x}+7\right)^2}, $$ or $$ -\frac{3}{y^4}\frac{dy}{dx}=-\frac{2e^{2x}}{\left(e^{2x}+7\right)^2}, $$ equivalently $$ \frac{d}{dx}\big(y^{-3}\big)=\frac{d}{dx}\left(\frac{1}{e^{2x}+7}\right), $$ and thus $$ y^{-3}=\frac{1}{e^{2x}+7}+c, $$ for some real constant $c$.

Answer 2

HINT:

So, we have $$\frac{dy}{y^4}=\frac{2e^{2x}dx}{3(e^{2x}+7)^2}$$

Integrate either sides by substitute $e^{2x}+7$ with $u$