# How do you solve this differential equation dy/dx=(-x)/y when y=3 and x=4 ?

Mar 13, 2018

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

#### Explanation:

We have:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- x}{y}$ with $y = 3$ when $x = 4$

This is a separable ODE, so we can write:

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

Then we can "separate the variables" :

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

Then we can readily integrate to get:

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

Given the initial condition $y \left(4\right) = 3$ then:

$\frac{1}{2} \cdot 9 = - \frac{1}{2} \cdot 16 + C \implies C = \frac{25}{2}$

So the Particular Solution is:

$\frac{1}{2} {y}^{2} = - \frac{1}{2} {x}^{2} + \frac{25}{2}$

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

Which we note is a circle of radius $5$ centred on the origin.