# What is a solution to the differential equation dy/dx=xy^2 with the particular solution y(2)=-2/5?

Jul 11, 2016

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

#### Explanation:

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

this is separable

$\frac{1}{y} ^ 2 \setminus \frac{\mathrm{dy}}{\mathrm{dx}} = x$

$\int \setminus \frac{1}{y} ^ 2 \setminus \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int \setminus x \setminus \mathrm{dx}$

$\int \setminus \frac{1}{y} ^ 2 \setminus \mathrm{dy} = \int \setminus x \setminus \mathrm{dx}$

power rule

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

$y \left(2\right) = - \frac{2}{5}$

$\frac{5}{2} = 2 + C \implies C = \frac{1}{2}$

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

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

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

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