To solve the given differential equation, proceed as follows:
[tex]\begin{gathered} \frac{dy}{dx}=\frac{2x}{e^{2y}} \\ e^{2y}dy=2xdx \end{gathered}[/tex]
Use integrals to solve for y and x:
[tex]\begin{gathered} \int e^{2y}dy=\int 2xdx \\ \frac{e^{2y}}{2}=x^2+C \end{gathered}[/tex]
Solve for y:
[tex]\begin{gathered} e^{2y}=2x^2+C \\ 2y=\ln |2x^2+C| \\ y=\frac{\ln |2x^2+C|}{2} \end{gathered}[/tex]
