# What is a solution to the differential equation dy/dx=(2x)/e^(2y)?

Dec 14, 2016

$y = \ln \sqrt{2 {x}^{2}}$

#### Explanation:

This is a separable differential equation.

$\mathrm{dy} \left({e}^{2 y}\right) = 2 x \mathrm{dx}$

$\int \left(\mathrm{dy} {e}^{2 y}\right) = \int \left(2 x \mathrm{dx}\right)$

$\frac{1}{2} {e}^{2 y} = {x}^{2}$

It is often preferable to solve for $y$.

${e}^{2 y} = {x}^{2} / \left(\frac{1}{2}\right)$

${e}^{2 y} = 2 {x}^{2}$

$\ln \left({e}^{2 y}\right) = \ln \left(2 {x}^{2}\right)$

$2 y \ln e = \ln \left(2 {x}^{2}\right)$

$2 y = \ln \left(2 {x}^{2}\right)$

$y = \frac{1}{2} \ln \left(2 {x}^{2}\right)$

$y = \ln \sqrt{2 {x}^{2}}$

Hopefully this helps!