Separable Variables of Differential Equations

The general differential equation of the first order and first degree
\[\frac{{dy}}{{dx}} = f\left( {x,y} \right)\,\,\,\,\,{\text{ – – – }}\left( {\text{i}} \right)\]

is said to have separable variables or is the separable variable differential equation if $$f\left( {x,y} \right)$$ can be expressed as a quotient (or product) of a function of $$x$$ only and a function of $$y$$ only, i.e.
\[f\left( {x,y} \right) = \frac{{g\left( x \right)}}{{h\left( y \right)}}\]
\[f\left( {x,y} \right) = \frac{{h\left( y \right)}}{{g\left( x \right)}}\]
\[f\left( {x,y} \right) = g\left( x \right)h\left( y \right)\]

If the variables are separable we may write the equation (i) in the form
\[\frac{{dy}}{{dx}} = \frac{{g\left( x \right)}}{{h\left( y \right)}}\,\,\,\,{\text{ – – – }}\left( {{\text{ii}}} \right)\]
\[g\left( x \right)dx – h\left( y \right)dy = 0\,\,\,\,{\text{ – – – }}\left( {{\text{iii}}} \right)\]

It is clear that when the variables are separated the solution of the differential equation is merely an exercise in integration. Thus the solution of (i) is obtained by integrating (ii) and (iii), i.e.
\[\int {g\left( x \right)dx – \int {h\left( y \right)dy = c} } \,\,\,\,{\text{ – – – }}\left( {{\text{iv}}} \right)\] where $$c$$ is the constant of integration.

Upon solving this equation for $$y$$ in terms of $$x$$, we have $$y = y\left( {x,c} \right)$$. Note that the constant of integration $$c$$ can be replaced by $$\ln c,\,{e^c},\,\sin c,\,{\tan ^{ – 1}}c$$ for simplification.