Second-Order Linear DEs

A second-order linear differential equation is an equation involving an unknown function and its first and second derivatives, linked by constant (or variable) coefficients. The most important form you will encounter in high school calculus is the homogeneous constant-coefficient equation: d²x/dt² + a·(dx/dt) + b·x = 0, where a and b are real constants. This single equation governs a remarkable variety of physical phenomena — the oscillation of a spring-mass system, the swing of a pendulum, the charge on a capacitor in an electrical circuit, and the decay of vibrations in a damping medium. What makes second-order equations special is that their general solution contains two arbitrary constants, reflecting the fact that specifying both an initial position and an initial velocity is required to uniquely determine future motion. This is physically natural: knowing where something is at time t = 0 is not enough; you also need to know how fast it is moving. The two constants are determined by two initial conditions, x(0) = x₀ and x'(0) = v₀. There are two powerful complementary approaches to solving these equations. The analytic (exact) approach uses the characteristic equation to find exponential or oscillatory closed-form solutions. The numerical approach — most accessibly Euler's method on a reduced first-order system — allows approximate solutions when exact formulas are difficult or impossible. Both methods are deeply connected: understanding the exact solution tells us what numerical methods must reproduce, and numerical experiments build intuition about qualitative behavior such as damping, resonance, and oscillation.

Key takeaways
  • The Standard Form
  • Reduction to a Coupled First-Order System
  • Applying Euler's Method to the System
  • The Characteristic Equation (Exact Solution)
  • Three Cases Based on the Discriminant Δ = a² − 4b
  • Applying Initial Conditions

The Standard Form

Write the equation in standard form: d²x/dt² + a·(dx/dt) + b·x = 0. The coefficient of d²x/dt² must be 1 (divide through if needed). Here a is the damping coefficient and b is the restoring coefficient. When a = 0 and b > 0 the system oscillates forever (simple harmonic motion). When a > 0 the system loses energy over time (damped oscillation or overdamped decay).

Reduction to a Coupled First-Order System

Introduce two new variables: let x₁ = x (position) and x₂ = dx/dt (velocity). Then dx₁/dt = x₂ and dx₂/dt = d²x/dt² = −a·x₂ − b·x₁. The original single second-order equation becomes the coupled system: dx₁/dt = x₂ and dx₂/dt = −b·x₁ − a·x₂. This is crucial: any numerical method (Euler, Runge-Kutta) operates on first-order systems, so this reduction is the bridge between theory and computation.

Applying Euler's Method to the System

With step size h, starting from (x₁⁰, x₂⁰) = (x₀, v₀), repeatedly apply: x₁^(n+1) = x₁ⁿ + h·x₂ⁿ and x₂^(n+1) = x₂ⁿ + h·(−b·x₁ⁿ − a·x₂ⁿ). Each step advances the solution by time h. Smaller h gives more accuracy but requires more steps. Euler's method is first-order accurate (global error proportional to h), which means halving h roughly halves the error.

The Characteristic Equation (Exact Solution)

For exact solutions, substitute the trial function x(t) = e^(rt) into d²x/dt² + a·(dx/dt) + b·x = 0. Because (d/dt)e^(rt) = r·e^(rt) and (d²/dt²)e^(rt) = r²·e^(rt), the equation becomes e^(rt)·(r² + a·r + b) = 0. Since e^(rt) ≠ 0, we need r² + a·r + b = 0. This is the characteristic equation. Its two roots r₁ and r₂ (found via the quadratic formula: r = (−a ± √(a² − 4b)) / 2) determine the nature of the solution.

Three Cases Based on the Discriminant Δ = a² − 4b

Case 1 — Overdamped (Δ > 0): Two distinct real roots r₁, r₂. General solution: x(t) = C₁·e^(r₁t) + C₂·e^(r₂t). The system decays without oscillating. Case 2 — Critically Damped (Δ = 0): One repeated root r = −a/2. General solution: x(t) = (C₁ + C₂·t)·e^(rt). This is the fastest decay without oscillation. Case 3 — Underdamped (Δ < 0): Complex roots r = α ± iω where α = −a/2 and ω = √(4b − a²)/2. General solution: x(t) = e^(αt)·(C₁·cos(ωt) + C₂·sin(ωt)). The system oscillates with exponentially decaying amplitude.

Applying Initial Conditions

Once the general solution is written with constants C₁ and C₂, substitute t = 0 and use x(0) = x₀ to get one equation, then differentiate x(t), substitute t = 0, and use x'(0) = v₀ to get a second equation. Solve these two simultaneous equations for C₁ and C₂. This gives the unique particular solution satisfying the given initial conditions.

The Key Equation

d2xdt2+adxdt+bx=0,r2+ar+b=0,r=−a±a2−4b2\frac{d^2x}{dt^2} + a\frac{dx}{dt} + bx = 0, \qquad r^2 + ar + b = 0, \qquad r = \frac{-a \pm \sqrt{a^2 - 4b}}{2}
  • x(t) — Dependent variable (varies (metres, coulombs, etc.)): The unknown function we are solving for — it could represent position, charge, or any quantity governed by this equation.
  • a — Damping coefficient (s⁻¹): Multiplies the first derivative; controls energy dissipation. When a > 0 the system is damped; when a = 0 oscillations persist forever.
  • b — Restoring coefficient (s⁻²): Multiplies x; controls the restoring force. For a spring, b = k/m where k is the spring constant and m is mass.
  • r — Characteristic root (dimensionless (or s⁻¹ in physics context)): The exponent in the trial solution x = e^(rt). Its sign and whether it is real or complex determines oscillatory versus decaying behavior.
  • \Delta = a^2 - 4b — Discriminant (s⁻²): Determines the nature of solutions: positive gives two real roots (overdamped), zero gives a repeated root (critically damped), negative gives complex roots (underdamped oscillation).

Worked Example

Problem. Solve d²x/dt² + 2·(dx/dt) + 5·x = 0 with initial conditions x(0) = 3 and x'(0) = 1. Then perform two steps of Euler's method with h = 0.1 on the equivalent first-order system and compare with the exact values.

x(t)=e−t(3cos⁡(2t)+2sin⁡(2t))x(t) = e^{-t}\bigl(3\cos(2t) + 2\sin(2t)\bigr)

Answer. Exact solution: x(t) = e^(−t)·(3·cos(2t) + 2·sin(2t)). Euler approximation at t = 0.2: x ≈ 3.03 vs exact ≈ 2.901.

Figure
Diagram
Phase Plane and Time-Domain: Underdamped, Critically Damped, OverdampedtxUnderdampedOverdampedCriticallyx₁x₂Phase Portrait (x vs x')Spiral inEquilibriumLeft: x(t) vs time for three damping cases. Right: phase portrait for underdamped spiral.
Origins · 1 minLeonhard Euler and how this was discovered
LE
Leonhard Euler
1707–1783 · Swiss

Euler invented the method of solving linear differential equations using exponential trial solutions (the characteristic equation approach), introduced the systematic study of second-order ODEs, and formulated the numerical stepping method now universally known as Euler's method. His notation e^(it) = cos(t) + i·sin(t), known as Euler's formula, is the cornerstone of the complex-roots case of the characteristic equation. Euler lost sight in his right eye at age 31 and became almost completely blind later in life — yet he produced roughly half of his total mathematical output after losing his full vision, dictating work to his sons and assistants.

  • First-Order Differential Equations — Second-order equations are reduced to coupled first-order systems before applying Euler's method; mastering first-order ODEs is a prerequisite.
  • Euler's Method — The primary numerical technique applied to the reduced first-order system to approximate solutions of second-order equations.
  • Complex Numbers — The underdamped case produces complex characteristic roots; Euler's formula converts these into real oscillatory solutions using sine and cosine.
  • The Quadratic Formula — The characteristic equation r² + ar + b = 0 is solved by the quadratic formula; its discriminant determines which of the three solution cases applies.
  • Simple Harmonic Motion — The undamped second-order equation d²x/dt² + ω²x = 0 is the mathematical model of simple harmonic motion, the special case with a = 0.