# What is a general solution to the differential equation y'=2+2x^2+y+x^2y?

Jul 24, 2016

$y = C {e}^{x + {x}^{3} / 3} - 2$

#### Explanation:

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

this is separable,

$y ' = 2 \left(1 + {x}^{2}\right) + y \left(1 + {x}^{2}\right)$

$y ' = \left(2 + y\right) \left(1 + {x}^{2}\right)$

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

$\int \setminus \frac{1}{2 + y} y ' \setminus \mathrm{dx} = \int \setminus \left(1 + {x}^{2}\right) \setminus \mathrm{dx}$

$\int \setminus \frac{d}{\mathrm{dx}} \left(\ln \left(2 + y\right)\right) \setminus \mathrm{dx} = \int \setminus \left(1 + {x}^{2}\right) \setminus \mathrm{dx}$

$\ln \left(2 + y\right) = x + {x}^{3} / 3 + C$

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

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

$y = C {e}^{x + {x}^{3} / 3} - 2$