first-order linear equation

A first-order linear equation is the second great solvable family, and 'linear' here has a precise, modest meaning: the unknown y and its derivative y' appear only to the first power and are never multiplied together, squared, or fed into a function like sin or exp. Picture an equation where y and y' show up plainly, each with at most a coefficient that may depend on x — no y^2, no y y', no sin(y).

The standard shape is y' + p(x) y = q(x). The coefficient p(x) and the right-hand side q(x) are known functions of x alone (they may be constants, or genuine functions of x). The term q(x) is the forcing or input; when q(x) = 0 the equation is called homogeneous and is actually separable, while a nonzero q makes it nonhomogeneous. The reason this family is so friendly is that it has a guaranteed solution method — the integrating factor — that works for ANY continuous p and q, not just lucky special cases. You are never left guessing.

Linearity is the property that powers almost all of elementary ODE theory, so it is worth fixing the idea early. Linear first-order equations model RC and RL circuits, mixing tanks, and Newton cooling with a time-varying environment, and their solutions have a clean structure (a homogeneous part plus one particular part). The honest contrast: nonlinear first-order equations like y' = y^2 can blow up in finite time or fail to have unique solutions, behaviour that linear equations, by their structure, simply cannot exhibit.