How do you find the general solution to dy/dx=(2x)/e^(2y)?

Jul 24, 2016

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

Explanation:

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

this is separable

${e}^{2 y} \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x$

$\int \setminus {e}^{2 y} \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int \setminus 2 x \setminus \mathrm{dx}$

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

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

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

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