Separable equations

Definition

Suppose that $f(y,t)=\dot{y}$ can be written in the form of $\frac{M(t)}{N(y)}$ for some $M,N$, then the differential equation

\frac{M(t)}{N(y)} = \frac{dy}{dt}

$$iscalleda\ast \ast separableequation\ast \ast .$$

One can solve such equation as following

$$\begin{array}{rl}\frac{M(t)}{N(y)}& =\frac{dy}{dt}\\ \int N(y)\frac{dy}{dt}dt& =\int M(t)dt\\ \int N(y)dy& =\int M(t)dt\end{array}$$