# What is a solution to the differential equation dy/dx=(1+x)(1+y)?

Jul 18, 2016

$y = C {e}^{x + {x}^{2} / 2} - 1$

#### Explanation:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \left(1 + x\right) \left(1 + y\right)$

this is separable!

$\frac{1}{1 + y} \frac{\mathrm{dy}}{\mathrm{dx}} = 1 + x$

so we integrate both sides

$\int \setminus \frac{1}{1 + y} \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int \setminus 1 + x \setminus \mathrm{dx}$

or

$\int \setminus \frac{1}{1 + y} \setminus \mathrm{dy} = \int \setminus 1 + x \setminus \mathrm{dx}$

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

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

$y = C {e}^{x + {x}^{2} / 2} - 1$