Separation of variables

A first-order ODE is separable if it can be written as the product of one function that depends only on the parameter $t$ and one that depends only on $y$:

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

We use separation of variables to solve ODEs of this class. The idea is that we put everything of one parameter on one side, and everything of another on the other, and then integrate. We only take one arbitrary constant of integration.

$$\int \frac{1}{h(y)}dy=\int g(t)dt$$

Note that this isn’t infallible. If the denominator of some resulting fraction is equal to 0, we must attempt a different method, often using the integrating factor.