constant-coefficient equation

Among all linear differential equations, the ones whose coefficients are plain numbers — not functions that change with x — are by far the easiest and the most common in elementary applications. A spring with a fixed stiffness, a circuit with fixed resistance and capacitance, radioactive decay with a fixed rate: each is modelled by a constant-coefficient equation, and each can be solved completely with a single algebraic shortcut.

A constant-coefficient second-order equation has the form a y'' + b y' + c y = g(x) where a, b, c are fixed real numbers (a not zero). In the homogeneous case g(x) = 0, the constancy of the coefficients is exactly what makes the exponential guess y = e^(r x) work: because differentiating e^(r x) just multiplies by the constant r, substituting reduces the whole equation to the algebraic characteristic equation a r^2 + b r + c = 0. Solve that quadratic, read its roots, and the three cases (distinct real, repeated, complex) deliver the general solution directly. No integration of variable coefficients, no series, no special functions — just algebra and exponentials.

This is why constant-coefficient equations are the first solvable class everyone learns and the bedrock of vibration and circuit theory: the systems engineers care about most often have constant parameters, at least to a first approximation. But the boundary is sharp. The moment a coefficient depends on x — say x y'' + y' + x y = 0 — the exponential trick collapses, e^(r x) is no longer a solution, and you must reach for entirely different tools (reduction of order, Cauchy-Euler substitutions, power series, or numerics). Constant coefficients are a special, blessed case, not the general rule.