# What is a solution to the differential equation dy/dx=-x/y with the particular solution y(1)=-sqrt2?

Sep 25, 2016

Multiply both sides by $y \mathrm{dx}$,integrate, and solve for the initial conditions

#### Explanation:

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

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

Because the form of C is arbitrary, we can write the above as our friend the circle:

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

Forcing the initial condition:

${1}^{2} + {\left(- \sqrt{2}\right)}^{2} = {C}^{2}$

${C}^{2} = 3$

The equation becomes:

${x}^{2} + {y}^{2} = {\left(\sqrt{3}\right)}^{2}$