Separation of variables

Separation of variables is a technique of solving a specific type of differential equations.

Definition

An ordainary differential equation is called separable if the following criteria can be satisfied through rearranging the equation:

$f\left(y\right)\,dy=g\left(x\right)\,dx$

To solve

$x\frac{dy}{dx}y=1$

We can rearrange the equation into

$y\,dy=\frac{1}{x}\,dx$

Integrating:

$\int y\,dy=\int \frac{1}{x}\,dx$

Which gives

$\frac{1}{2}y^2+C_1=\ln\left(x\right)+C_2$

Since $C 1$ and $C 2$ are arbitaray constants, we can group them together and give

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

The answer is usually leave at the form described above instead of isolating $y$ .

D. Lomen and D. Lovelock, Differential Equations Graphics. Model. Data., John Wiley and Sons, Toronto, 1999.
Separation of variables on Wolfram Mathworld