Separable Differential Equation

$$\frac{dy}{dt}=f(t)\cdot g(y)$$

Techniques

Separation of Variables

Given a separable differential equation, it is guaranteed that a n Implicit Solution formula can be derived. An Explicit Solution for $y$ may be possible.

$$\begin{array}{rl}\frac{dy}{dt}=& f(t)\cdot g(y)\\ \frac{1}{g(y)}dy=& f(t)dt\\ \int \frac{1}{g(y)}dy=& \int f(t)dt\end{array}$$

Example 1

$$\begin{array}{rl}\frac{dy}{dt}=& e^{2y}\cos (t)\\ \int e^{-2y}dy=& \int \cos (t)dt\\ -\frac{1}{2}{e}^{-2y}=& \sin (t)+C_1\leftarrow \text{implicit sol.}\\ e^{-2y}=& -2\sin (t)+C\\ y(t)=& -\frac{1}{2}\ln (-2\sin (t)+C)\leftarrow \text{explicit sol.}\end{array}$$