Second-Order Linear Homogeneous Equations

Step up one order. A second-order linear homogeneous equation with constant coefficients is

a\,y'' + b\,y' + c\,y = 0, \qquad a \ne 0.

"Homogeneous" means the right side is zero — no external forcing, just the system's own internal dynamics. These equations are the heartbeat of physics: every oscillation, every decaying vibration, every LC circuit obeys one. Unlike the first-order equations we met by separating or integrating factors, we solve this one by an inspired guess — and the guess turns the whole calculus problem into algebra.

The guess, and the characteristic equation

Step 1 — guess an exponential. What function reproduces itself under differentiation, so that y'', y' and y can cancel? The exponential. Try

y = e^{rx}, \qquad y' = r\,e^{rx}, \qquad y'' = r^2\,e^{rx}.

Step 2 — substitute into the ODE.

a\,r^2 e^{rx} + b\,r\,e^{rx} + c\,e^{rx} = 0.

Step 3 — factor out e^{rx}. It is never zero, so we may divide it away, and the differential equation becomes a plain quadratic — the characteristic equation:

e^{rx}\big(a r^2 + b r + c\big) = 0 \;\Longrightarrow\; \boxed{\,a r^2 + b r + c = 0.\,}

Calculus has dissolved into algebra. The roots r of this quadratic decide everything, and the quadratic formula sorts them into three cases by the sign of the discriminant \Delta = b^2 - 4ac. (Two independent solutions are needed, since the general solution of a second-order ODE carries two constants.)

Case 1 — distinct real roots (\Delta > 0)

Two different real roots r_1 \ne r_2 give two independent exponential solutions, and the general solution is their combination:

y = C_1 e^{r_1 x} + C_2 e^{r_2 x}.

Worked example. For y'' - y' - 6y = 0 the characteristic equation is r^2 - r - 6 = (r - 3)(r + 2) = 0, so r = 3, -2 and

y = C_1 e^{3x} + C_2 e^{-2x}.

Case 2 — a repeated root (\Delta = 0)

When \Delta = 0 the formula gives a single root r = -b/(2a), and e^{rx} is only one solution — we are short one. The missing partner is x\,e^{rx}: multiplying by x supplies the second independent solution (reduction of order confirms it). Hence

y = (C_1 + C_2 x)\,e^{rx}.

Worked example. For y'' - 4y' + 4y = 0 the characteristic equation is r^2 - 4r + 4 = (r - 2)^2 = 0, a repeated root r = 2, so

y = (C_1 + C_2 x)\,e^{2x}.

Case 3 — complex roots (\Delta < 0)

A negative discriminant gives a conjugate pair r = \alpha \pm i\beta. The solution e^{(\alpha \pm i\beta)x} is genuine but complex; Euler's formula e^{i\theta} = \cos\theta + i\sin\theta trades it for two real solutions, e^{\alpha x}\cos\beta x and e^{\alpha x}\sin\beta x. Hence

y = e^{\alpha x}\big(C_1 \cos\beta x + C_2 \sin\beta x\big).

The real part \alpha is an exponential envelope (growth or decay); the imaginary part \beta sets the oscillation frequency.

Worked example — simple harmonic motion. The purest oscillator is y'' + \omega^2 y = 0 (here a=1, b=0, c=\omega^2). Its characteristic equation is r^2 + \omega^2 = 0, with roots r = \pm i\omega — purely imaginary, so \alpha = 0, \beta = \omega — and

y = C_1 \cos\omega t + C_2 \sin\omega t.

No envelope, just an undamped sinusoid of angular frequency \omega — a frictionless mass on a spring, oscillating forever.

For a y'' + b y' + c y = 0, substitute y = e^{rx} to obtain the characteristic equation a r^2 + b r + c = 0. With discriminant \Delta = b^2 - 4ac, the general solution is:

Distinct real roots (\Delta > 0), r_1 \ne r_2: \;y = C_1 e^{r_1 x} + C_2 e^{r_2 x}.
Repeated root (\Delta = 0), r = -b/2a: \;y = (C_1 + C_2 x)\,e^{rx}.
Complex roots (\Delta < 0), r = \alpha \pm i\beta: \;y = e^{\alpha x}\big(C_1 \cos\beta x + C_2 \sin\beta x\big).

In each case the two arbitrary constants C_1, C_2 are fixed by two initial conditions, e.g. y(0) and y'(0).

Hang a mass m on a spring (stiffness k) in a medium that resists motion with damping c. Newton's second law gives the archetypal second-order equation

m\,y'' + c\,y' + k\,y = 0,

and the three root cases become three physical regimes, decided by how the damping c compares with the critical value c_c = 2\sqrt{mk}:

Underdamped (c < c_c, complex roots): the mass oscillates inside a shrinking envelope e^{\alpha t}, \alpha < 0 — a struck bell ringing down.
Critically damped (c = c_c, repeated root): the fastest return to rest without overshooting — the regime a car's suspension or a door closer is tuned to.
Overdamped (c > c_c, distinct real roots, both negative): a sluggish, non-oscillating crawl back to equilibrium, like a hinge in thick grease.

Set c = 0 and you recover the undamped y'' + \omega^2 y = 0 with \omega = \sqrt{k/m} — simple harmonic motion forever. The discriminant of a quadratic, it turns out, is the difference between a bell that rings and a door that merely sighs shut.

Watch the regimes change, live

Below is the solution of the normalised oscillator y'' + 2\zeta\,y' + y = 0 started from y(0) = 1, y'(0) = 0, with the damping ratio \zeta on a slider (\zeta = c/c_c). The characteristic equation is r^2 + 2\zeta r + 1 = 0, so the discriminant is 4(\zeta^2 - 1): at \zeta < 1 you see the underdamped ringing, at \zeta = 1 the critical knife-edge, and at \zeta > 1 the overdamped crawl. The dashed envelope is \pm e^{-\zeta t}.