There are some more classifications for (second order) odes, mostly based on the fact that the ode is solvable (or not) when it is of a certain class. I like the choice of the Maple software, which is pretty classic. The online description of odeadvisor giving you a classification is here:
Your first order ode is homogeneous, because it does not have a term that only depends on t. dx/dt=a(t)*x+b(t) is not homogeneous, but dx/dt = a(t)*x is. Your example is also separable, which means it can be solved using separation of variables.