# What is a solution to the differential equation y'=xe^(x^2-lny^2)?

Aug 29, 2016

$y = \sqrt[3]{\frac{3}{2} {e}^{{x}^{2}} + C}$

#### Explanation:

$y ' = x {e}^{{x}^{2} - \ln {y}^{2}}$

we can separate it out first

$y ' = x {e}^{{x}^{2}} {e}^{- \ln {y}^{2}}$

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

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

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

$\int {y}^{2} y ' \mathrm{dx} = \int x {e}^{{x}^{2}} \mathrm{dx}$

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

.....spotting the pattern on the RHS

${y}^{3} / 3 = \int \frac{d}{\mathrm{dx}} \left(\frac{1}{2} {e}^{{x}^{2}}\right) \mathrm{dx}$

${y}^{3} / 3 = \frac{1}{2} {e}^{{x}^{2}} + C$

${y}^{3} = \frac{3}{2} {e}^{{x}^{2}} + C$

$y = \sqrt[3]{\frac{3}{2} {e}^{{x}^{2}} + C}$