Solving a Separable First-Order Differential Equation

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 \(y.\) \[ \begin{eqnarray*} y^{3} &=&6e^{x}+3C \\ y &=&\sqrt[3]{6e^{x}+3C} \end{eqnarray*} \]