separable equation
The separable equation is the friendliest first-order equation there is, and the one you should always test for first. The lucky structure is that the two variables can be peeled apart — everything involving y herded to one side, everything involving x herded to the other — so that the differential equation collapses into two ordinary integrals you already know how to do. When it works, solving is almost mechanical.
An equation is separable if dy/dx can be written as a product g(y)*h(x), one factor in y alone and one in x alone. Then you separate: move it to (1/g(y)) dy = h(x) dx, and integrate both sides independently — integral of (1/g(y)) dy = integral of h(x) dx + C. The single constant C from the two integrations carries the family. For dy/dx = x*y, dividing by y gives (1/y) dy = x dx; integrating, ln|y| = x^2/2 + C, so y = A*e^(x^2/2). The whole method is just the chain rule run in reverse, dressed up.
Separable equations are everywhere in elementary modeling because so many basic laws say 'the rate is proportional to something simple': radioactive decay, Newton's law of cooling, unconstrained population growth, and (after the first algebraic step) the logistic equation all separate. The honest limitation is that separability is a structural accident — most equations are not separable, and you cannot force them to be. The two follow-up tricks for that are the integrating factor for linear equations and substitutions for the homogeneous and Bernoulli forms.
Newton's law of cooling, dT/dt = -k*(T - T_env), separates into dT/(T - T_env) = -k dt. Integrating both sides gives ln|T - T_env| = -k*t + C, so T = T_env + A*e^(-k*t): the temperature relaxes exponentially to the room.
Separate the variables, integrate each side once, and the family appears.
When you divide by g(y) to separate, you may quietly throw away the constant solutions where g(y) = 0 (here T = T_env). These are real solutions and should be checked separately — division by a factor that can vanish is the classic place separable methods lose a solution.