Solve $\dfrac{dy}{dx}=\dfrac{2e^{x}}{y^{2}}$

Solution Use the steps from above:

Step 1 Express $\dfrac{dy}{dx}=\dfrac{2e^{x}}{y^{2}}$ in the differential form: $$y^{2}dy-2e^{x}dx=0\qquad \hbox{or equivalently, as}\qquad y^{2}dy=2e^{x}dx$$

Step 2 Integrate to obtain the general solution. $$\begin{eqnarray*} \int y^{2}dy &=&\int 2e^{x}dx \\ \dfrac{y^{3}}{3} &=&2e^{x}+C \end{eqnarray*}$$

where $C$ is a constant. This solution is expressed implicitly. To obtain the explicit form, solve for $y.$ $$\begin{eqnarray*} y^{3} &=&6e^{x}+3C \\ y &=&\sqrt[3]{6e^{x}+3C} \end{eqnarray*}$$