Separation of variables

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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 C1 and C2 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
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