Separable ODEs

A separable ODE is one in which the functions of both x and y can be separated.

$$\frac{dy}{dx} = \frac{g(x)}{f(y)}$$

$$\frac{dy}{dx} f(y) = g(x)$$

This can then be rearranged and integrated so that

$$\int f(y) dy = \int g(x) dx\\$$

Example

$$\frac{dy}{dx} = e^{x+y} = e^xe^y$$

$$\int e^{-y} dy = \int e^x dx$$

$$-e^{-y} = e^x + c$$

$$e^x + e^{-y} = c$$

$c$ is just a constant so its sign does not matter at this point.
We can then solve explicitly for $y$.

$$e^{-y} = c - e^x$$

$$y = -\ln(c-e^x)$$