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:

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

Example

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 $y$.

Reference

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