Variable separable equations

Variable separable differential equations are of the form \(\frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}(x)\mathrm{g}(y)\).

If the differential equation is of this form, then we can rearrange:

\[\frac{1}{\mathrm{g}(y)} \frac{\mathrm{d}y}{\mathrm{d}x} = \mathrm{f}(x)\]

\[\Rightarrow \int \frac{1}{\mathrm{g}(y)} \frac{\mathrm{d}y}{\mathrm{d}x} \, \mathrm{d}x = \int \mathrm{f}(x) \, \mathrm{d}x\]

\[\Rightarrow \int \frac{1}{\mathrm{g}(y)} \, \mathrm{d}y = \int \mathrm{f}(x) \, \mathrm{d}x \quad \text{(by chain rule)}\]