# What is a particular solution to the differential equation dy/dx=-x/y with y(4)=3?

Jul 18, 2016

$y = \pm \sqrt{25 - {x}^{2}}$

#### Explanation:

equation is separable as follows

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

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

$\int \setminus y \setminus \mathrm{dy} = - \int \setminus x \setminus \mathrm{dx}$

${y}^{2} / 2 = - \left({x}^{2} / 2 - C\right)$

${y}^{2} / 2 = C - {x}^{2} / 2$

${y}^{2} = C - {x}^{2}$

applying the IV

$9 = C - {4}^{2} , \implies C = 19$

${y}^{2} = 25 - {x}^{2}$

$y = \pm \sqrt{25 - {x}^{2}}$