Linear ODEs with Constant Coefficients Easy Explanation

A linear ordinary differential equation with constant coefficients looks like ay″ + by′ + cy = f(x), where a, b, c are just numbers. These equations run the physical world: the mass-spring oscillator, the RLC circuit, a damped pendulum, and countless control systems all obey them. The magic is that constant coefficients make the equation solvable by pure algebra — you guess an exponential, and the differential equation collapses into a polynomial called the characteristic equation.

This matters because "guess e^{rx}" converts calculus into root-finding. The three possible root patterns — two distinct real roots, one repeated root, or a complex pair — give the three great behaviours of physics: exponential decay or growth, critically damped return, and oscillation. The method of undetermined coefficients then handles forcing terms f(x), producing resonance phenomena like a bridge swaying in the wind.

Key Concepts

• Order and linearity: the order is the highest derivative; linear means y and its derivatives appear only to the first power, never multiplied together.
• Homogeneous vs nonhomogeneous: ay″ + by′ + cy = 0 is homogeneous (solution: complementary function y_c); with f(x) ≠ 0 on the right it is nonhomogeneous (add a particular solution y_p).
• Characteristic equation: substituting y = e^{rx} into ay″ + by′ + cy = 0 gives ar² + br + c = 0. Its roots dictate the entire solution.
• Three root cases: distinct real r₁, r₂ → y = C₁e^{r₁x} + C₂e^{r₂x}; repeated r → y = (C₁ + C₂x)e^{rx}; complex α ± iβ → y = e^{αx}(C₁cosβx + C₂sinβx).
• Superposition: sums of solutions are solutions (homogeneous case), so initial conditions fix the constants C₁, C₂ uniquely.

Formulas

• Standard form: y″ + py′ + qy = f(x), with p, q constants. Characteristic equation: r² + pr + q = 0.
• Distinct real roots: y_c = C₁e^{r₁x} + C₂e^{r₂x}.
• Repeated root r: y_c = (C₁ + C₂x)e^{rx} — the extra x is essential, not optional.
• Complex roots α ± iβ: y_c = e^{αx}(C₁cosβx + C₂sinβx), describing oscillations of frequency β with envelope e^{αx}.
• Undetermined coefficients: for polynomial, exponential, or sinusoidal f(x), guess y_p of a similar form (times x^s if it duplicates y_c, where s is the multiplicity).

Worked Example

Solve y″ − 3y′ + 2y = 0 with y(0) = 1 and y′(0) = 0. The characteristic equation is r² − 3r + 2 = 0, factoring as (r − 1)(r − 2) = 0, so r = 1 and r = 2 (distinct real roots). The general solution is y = C₁e^x + C₂e^{2x}. Apply y(0) = 1: C₁ + C₂ = 1. Differentiate: y′ = C₁e^x + 2C₂e^{2x}, and y′(0) = 0 gives C₁ + 2C₂ = 0. Subtracting: C₂ = −1, then C₁ = 2. The solution is y = 2e^x − e^{2x}. Verify: y′ = 2e^x − 2e^{2x}, y″ = 2e^x − 4e^{2x}; then y″ − 3y′ + 2y = (2e^x − 4e^{2x}) − 3(2e^x − 2e^{2x}) + 2(2e^x − e^{2x}) = (2 − 6 + 4)e^x + (−4 + 6 − 2)e^{2x} = 0. As a forced example, y″ + 4y = 3cos(2x) needs y_p = Ax sin(2x) (resonance: cos2x duplicates y_c), giving A = 3/4, so y_p = (3x/4)sin(2x) — amplitude growing linearly, the signature of resonance.

Common Misconceptions

1. "The method works for variable coefficients." No — e^{rx} only reduces the equation to algebra when the coefficients are constant. Variable coefficients need series or special-function methods.
2. "The repeated-root solution is just C₁e^{rx}." That gives only one constant for a second-order equation. The second independent solution is xe^{rx} — forgetting it loses half the solutions.
3. "Initial conditions apply only to the homogeneous part." They apply to the full general solution y = y_c + y_p after everything is assembled.

Real-World Applications

• Mass-spring-damper systems: mx″ + cx′ + kx = 0 classifies suspension as underdamped, critically damped, or overdamped from the root pattern.
• RLC circuits: LQ″ + RQ′ + Q/C = V(t) governs radios, filters, and power supplies; resonance tunes the station.
• Control engineering: the characteristic roots (poles) of a feedback loop determine stability — right-half-plane roots mean runaway.
• Beam deflection: the Euler-Bernoulli beam equation with constant cross-section reduces to a constant-coefficient ODE.

Constant coefficients turn differential equations into algebra: find the characteristic roots, write down the matching exponentials or oscillations, fit the constants to the initial conditions. Three root patterns cover nearly every vibration, circuit, and control problem in classical physics and engineering.