pure-time equation y' = f(x)
The simplest differential equation of all is one where the right side does not mention the unknown at all. A pure-time equation has the form y' = f(x): the rate of change depends only on the independent variable x (often time), not on y itself. It barely deserves the name 'differential equation' — it is really just an integration problem wearing a derivative's clothes.
Because the right side is a known function of x alone, there is nothing to disentangle. You solve it by direct integration: y = integral of f(x) dx + C, the antiderivative of f plus an arbitrary constant. Both the separable method and the linear method (with p(x) = 0) reduce to this same single step here; the integrating factor is just mu = e^0 = 1, so nothing is multiplied. The arbitrary constant C is exactly the value the solution would have at a chosen starting point, set by an initial condition. The name 'pure-time' comes from the picture where x is time: you are told the velocity f(t) at every instant and you recover position by accumulating it.
This case matters as the base camp from which the harder cases depart, and it makes the role of the constant of integration concrete. It also delivers an honest reminder: even this easiest equation may have no elementary answer, because integral of f(x) dx need not be an elementary function. The classic case is y' = e^(-x^2), whose solution involves the error function — no formula in elementary terms exists, even though the equation could hardly be simpler.
y' = 6x with y(0) = 2 integrates directly: y = 3x^2 + C. The condition y(0) = 2 gives C = 2, so y = 3x^2 + 2. No separation or integrating factor is needed — just one antiderivative.
When y' depends on x only, solving is pure integration and the constant is set by the initial value.
Do not over-think this case — there is no y on the right, so no separation, no integrating factor, no homogeneous part. But 'integrable in principle' is not 'has an elementary formula': y' = sin(x^2) needs a special function.