 # How to Write and Solve Separable Differential Equations

A separable differential equation is a differential equation where the variables can be separated to either side of the equal sign. This means that you can put all the $x$-terms on one side and all the $y$-terms on the other. It is best to collect all the $y$-terms on the left-hand side and all the $x$-terms on the right-hand side. Generally, you write separable equations on this form:

Rule

### SeparableDifferentialEquations

$\begin{array}{llll}\hfill & g\left(y\right){y}^{\prime }=f\left(x\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ⇒\int g\left(y\right)\phantom{\rule{0.17em}{0ex}}dy=\int f\left(x\right)\phantom{\rule{0.17em}{0ex}}dx+C\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $g\left(y\right){y}^{\prime }=f\left(x\right)⇒\int g\left(y\right)\phantom{\rule{0.17em}{0ex}}dy=\int f\left(x\right)\phantom{\rule{0.17em}{0ex}}dx+C$

Example 1

Solve the differential equation $2xy{y}^{\prime }={x}^{2}$

$\begin{array}{llll}\hfill 2xy{y}^{\prime }& ={x}^{2}\phantom{\rule{2em}{0ex}}|÷2x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y{y}^{\prime }& =\frac{{x}^{2}}{2x}=\frac{x}{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \int y\phantom{\rule{0.17em}{0ex}}dy& =\int \frac{x}{2}\phantom{\rule{0.17em}{0ex}}dx\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{1}{2}{y}^{2}& =\frac{{x}^{2}}{4}+C\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill y& =±\sqrt{\frac{{x}^{2}}{2}+C}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$