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:

<math>f\left(y\right)\,dy=g\left(x\right)\,dx</math>

Example

To solve

<math>x\frac{dy}{dx}y=1</math>

We can rearrange the equation into

<math>y\,dy=\frac{1}{x}\,dx</math>

Integrating:

<math>\int y\,dy=\int \frac{1}{x}\,dx</math>

Which gives

<math>\frac{1}{2}y^2+C_1=\ln\left(x\right)+C_2</math>

Since <math>C_1</math> and <math>C_2</math> are arbitaray constants, we can group them together and give

<math>\frac{1}{2}y^2=\ln\left(x\right)+C</math>

The answer is usually leave at the form described above instead of isolating <math>y</math>.

Reference

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