# How to you find the general solution of dy/dx=x/y?

Jan 13, 2017

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

#### Explanation:

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

$y \mathrm{dy} = x \mathrm{dx}$ by exploiting the notation (separation)

$\int y \mathrm{dy} = \int x \mathrm{dx}$ further exploiting the notation

$\frac{1}{2} {y}^{2} = \frac{1}{2} {x}^{2} + d$
${y}^{2} = {x}^{2} + 2 d$
${x}^{2} - {y}^{2} = - 2 d$
${x}^{2} - {y}^{2} = c$ where $c = - 2 d$
Depending on whether $c$ is positive, negative or zero you get a hyperbola open to the $x$-axis, open to the $y$=axis, or a pair of straight lines through the origin.