# What is a general solution to the differential equation dy/dx=xe^-y?

Jul 19, 2016

This is a separable differential equation.

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

$\frac{\mathrm{dy}}{\mathrm{dx}} = x \cdot \frac{1}{{e}^{y}}$

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

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

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

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

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