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: a mass bobbing on a spring,
a swinging pendulum (for small swings), the current sloshing in an
LC circuit — anything left to its own devices that either oscillates or settles
obeys an equation of exactly this shape. 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}.
A second example. For y'' - 5y' + 6y = 0 the substitution
y = e^{rx} gives r^2 - 5r + 6 = (r-2)(r-3) = 0,
so r = 2, 3 and
y = C_1 e^{2x} + C_2 e^{3x}.
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}.
A second example. For y'' + 4y' + 4y = 0 the
characteristic equation is r^2 + 4r + 4 = (r+2)^2 = 0, a repeated root
r = -2. Forgetting the extra factor of x would
leave only y = C_1 e^{-2x} — a one-parameter family, one constant short of
the two a second-order equation needs. Multiplying by x restores the
missing degree of freedom:
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.
A concrete second example. For y'' + 4y = 0 the
characteristic equation is r^2 + 4 = 0, giving
r = \pm\sqrt{-4} = \pm 2i. Read that root straight off:
\alpha = 0 (no growth or decay, the real part is zero) and
\beta = 2 (the imaginary part), so
y = C_1\cos 2x + C_2 \sin 2x,
a pure oscillation of angular frequency \omega = \beta = 2 — the very
number sitting inside the imaginary part of the root, with period 2\pi/2 = \pi.
Whenever the characteristic roots are purely imaginary, you can read the oscillation's frequency
directly off them.
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.
A plucked guitar string lives in the same family: tension and mass per length
play the roles of k and m, and a tiny amount
of air and internal friction supplies a small c. The string is heavily
underdamped, so the complex-root case wins — and the imaginary part \beta
of the root is literally the pitch you hear. A car's shock absorber, tuned instead to sit
right at (or just past) critical damping, is engineered to land in the other case entirely: it
must never ring like a plucked string, only settle.
-
Get the substitution table right. y'' becomes
r^2, y' becomes r
(not r^2!), and plain y becomes
1 (not r). Mixing up which power of
r goes with which derivative is the single most common slip in this
whole topic.
-
A repeated root needs the extra factor of x, or you silently
lose a solution. Writing y = C_1 e^{rx} + C_2 e^{rx} for a
repeated root is really just y = (C_1 + C_2)e^{rx} — one constant in
disguise, not two, and it cannot satisfy two independent initial conditions.
-
Complex roots always arrive in conjugate pairs. Because
a, b, c are real, a root \alpha + i\beta
forces its partner to be exactly \alpha - i\beta — never some other
complex number. If your quadratic formula ever hands back an "unpaired" complex root, go back
and check the arithmetic.
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}.
See it explained