Second-Order Linear Homogeneous Differential Equations

Core Idea

A second-order linear homogeneous ODE has the form y'' + p(x)y' + q(x)y = 0. The general solution is a linear combination of two linearly independent solutions: y = c₁y₁ + c₂y₂. For constant coefficients, solutions have exponential or trigonometric form determined by a characteristic equation. These equations model vibrations, electrical circuits, and countless physical phenomena.

Explainer

You already know how to solve first-order linear ODEs, which have a one-parameter family of solutions y = Ce^{∫−p dx}. Second-order equations are one level harder: y'' + p(x)y' + q(x)y = 0 involves two derivatives, and the general solution has two free constants c₁ and c₂. Two constants are needed because specifying a unique solution requires two initial conditions — typically the value of y and the value of y' at a single point.

The structure of the general solution is y = c₁y₁ + c₂y₂, where y₁ and y₂ are any two linearly independent solutions. "Linearly independent" means neither is a constant multiple of the other — they represent genuinely different behaviors of the system. Any solution you could possibly find is a linear combination of these two, so once you have y₁ and y₂, you have everything. This is why the prerequisite on linear independence matters: it gives you the language to confirm you have a complete solution set.

For constant-coefficient equations y'' + py' + qy = 0, the key insight is the trial solution y = eʳˣ. Since (eʳˣ)'' = r²eʳˣ and (eʳˣ)' = reʳˣ, substituting into the ODE gives (r² + pr + q)eʳˣ = 0. Since eʳˣ ≠ 0, this reduces to the characteristic equation r² + pr + q = 0 — a plain quadratic. Its roots determine the solution form:

• Two distinct real roots r₁, r₂: y = c₁e^r₁x + c₂e^r₂x
• Repeated root r: y = (c₁ + c₂x)eʳˣ
• Complex conjugate roots α ± βi: y = eᵅˣ(c₁cos(βx) + c₂sin(βx))

The complex-root case explains why these equations model oscillation. A simple spring-mass system satisfies y'' + ky = 0 (no damping term), whose characteristic roots are ±i√k — pure imaginary. The solution is c₁cos(√k · x) + c₂sin(√k · x), perpetual sinusoidal motion. Adding a damping term y' shifts the roots into the left complex half-plane, introducing a decaying exponential factor eᵅˣ with α < 0: the system oscillates but with amplitude that shrinks over time. The characteristic equation encodes all of this physics in two numbers.