Higher-Order Linear Equations

The second-order theory extends to order $n$ with no essential change. An $n$ -th order linear equation has the same structure: an $n$ -dimensional solution space, a Wronskian test for a basis, a characteristic polynomial whose roots build the general solution, and undetermined-coefficients and variation-of-parameters methods for forcing. The one new phenomenon is a root of multiplicity greater than two. Coupled oscillators are the physical setting that produces equations of order four and higher.

The general theory at order n

An $n$ -th order linear equation, after dividing by the leading coefficient, is

L [y] = y^{(n)} + p_{1} (t) y^{(n - 1)} + \dots + p_{n - 1} (t) y^{'} + p_{n} (t) y = g (t),

with the $p_{i}$ and $g$ continuous on an interval $I$ . Solving it should require $n$ integrations, each adding one constant, so an initial value problem now fixes the function and its first $n - 1$ derivatives at a point.

The homogeneous equation $L [y] = 0$ has an $n$ -dimensional solution space. Any $n$ solutions $y_{1}, \dots, y_{n}$ span it exactly when their Wronskian, the determinant of the functions and their first $n - 1$ derivatives,

W [y_{1}, \dots, y_{n}] = y_{1} y_{1}^{'} ⋮ y_{1}^{(n - 1)} y_{2} y_{2}^{'} ⋮ y_{2}^{(n - 1)} \dots \dots \dots y_{n} y_{n}^{'} ⋮ y_{n}^{(n - 1)},

is nonzero. As in the second-order case, this Wronskian is either identically zero or never zero on $I$ , so a single-point test decides it.

Such a set is a fundamental set of solutions and its combination is the general solution. Linear independence means no nontrivial constants satisfy $k_{1} y_{1} + \dots + k_{n} y_{n} \equiv 0$ ; for arbitrary functions this does not force a nonzero Wronskian, but for solutions of a common linear equation it does, which is what makes the Wronskian a reliable test here.

The homogeneous solution space at order

n

n

-dimensional; any

n

solutions with nonzero Wronskian form a basis for it.

Constant coefficients: the characteristic polynomial

For constant coefficients $a_{0} y^{(n)} + a_{1} y^{(n - 1)} + \dots + a_{n} y = 0$ , the substitution $y = e^{r t}$ again works, giving $L [e^{r t}] = e^{r t} Z (r)$ with the characteristic polynomial

Z (r) = a_{0} r^{n} + a_{1} r^{n - 1} + \dots + a_{n - 1} r + a_{n} .

By the fundamental theorem of algebra $Z$ has $n$ roots (with multiplicity), possibly repeated or complex, and factors as

Z (r) = a_{0} (r - r_{1}) (r - r_{2}) \dots (r - r_{n}) .

Each root contributes solutions by the same rules established for second order, now allowing higher multiplicity.

The degree-

n

characteristic polynomial factors into root blocks; each block, by its multiplicity and type, contributes basis solutions.

Complex roots come in conjugate pairs (the coefficients are real), each pair $λ \pm i μ$ contributing $e^{λ t} cos μ t$ and $e^{λ t} sin μ t$ , just as at second order. A fourth-order equation can mix real and complex roots.

Roots of

r^{4} - 1 = 0

in the complex plane: a real pair on the axis (one growing, one decaying) and an imaginary pair giving

cos t, sin t

Repeated roots and multiplicity

The one new feature is that a root can now repeat more than twice. If $r$ is a root of $Z$ of multiplicity $s$ (so $(r - r_{1})^{s}$ divides $Z$ ), it contributes $s$ independent solutions

e^{r t}, t e^{r t}, t^{2} e^{r t}, \dots, t^{s - 1} e^{r t} .

The rule generalizes the second-order repeated root ${e^{r t}, t e^{r t}}$ : multiplicity $s$ multiplies the exponential by $1, t, \dots, t^{s - 1}$ . A repeated complex pair $λ \pm i μ$ of multiplicity $s$ contributes $2 s$ solutions, the sines and cosines each multiplied by $1, t, \dots, t^{s - 1}$ .

The table collects the contribution of each root type to the basis.

Root of $Z (r)$	Multiplicity	Contribution to the basis
real $r$	$1$	$e^{r t}$
real $r$	$s$	$e^{r t}, t e^{r t}, \dots, t^{s - 1} e^{r t}$
complex $λ \pm i μ$	$1$	$e^{λ t} cos μ t, e^{λ t} sin μ t$
complex $λ \pm i μ$	$s$	$t^{j} e^{λ t} cos μ t, t^{j} e^{λ t} sin μ t$ , $j = 0, \dots, s - 1$

Summing the contributions of all roots always yields exactly $n$ independent solutions, matching the dimension of the solution space.

Nonhomogeneous higher-order equations

Both nonhomogeneous methods carry over. Undetermined coefficients uses the same trial forms, with one adjustment: because a characteristic root can now have multiplicity up to $n$ , a colliding trial may need to be multiplied by $t^{s}$ (rather than at most $t^{2}$ ) to clear all overlap with the homogeneous solution, where $s$ is the multiplicity of the offending root. Variation of parameters generalizes through the $n \times n$ Wronskian: the coefficient functions $u_{i}^{'}$ solve an $n \times n$ linear system built from the fundamental set and the forcing, then integrate. Reduction of order also applies but is rarely useful beyond second order, since it only lowers the order by one and the reduced equation is still of order $n - 1$ .

Coupled oscillators

Higher-order equations appear naturally when several oscillators are linked. Two masses connected by springs (each mass to a wall, and to each other) have two displacements $u_{1}, u_{2}$ obeying coupled second-order equations. Eliminating one variable produces a single fourth-order equation in the other, whose characteristic polynomial factors into two quadratic blocks. The roots are two pairs of imaginary numbers $\pm i ω_{a}$ and $\pm i ω_{b}$ , giving two normal-mode frequencies: the system oscillates as a superposition of two pure sinusoids, one where the masses move together and one where they move opposite.

Two masses coupled by springs; eliminating one displacement yields a fourth-order equation with two normal-mode frequencies.

The two normal modes: in the slow mode the masses move together, in the fast mode they move in opposition and compress the middle spring.

In general, a symmetric coupled system decouples into independent oscillators after the right change of variables, and each decoupled equation is a second-order harmonic oscillator. The change of variables that achieves this is exactly the diagonalization studied in the systems module.

Rather than eliminate variables to raise the order, the coupled equations can be kept as a first-order linear system in the vector $(u_{1}, u_{2}, u_{1}^{'}, u_{2}^{'})$ , where the normal modes reappear as eigenvectors of a matrix. The same reformulation turns any $n$ -th order equation into a first-order system.¹²³

Boyce, Elementary Differential Equations, §4.1 — General Theory of nth-Order Linear Equations: the existence-uniqueness theorem (4.1.1), the $n \times n$ Wronskian, and the equivalence of linear independence and fundamental sets (4.1.3). ↩
Boyce, Elementary Differential Equations, §4.2–§4.3 — Homogeneous Equations with Constant Coefficients and Undetermined Coefficients: the degree- $n$ characteristic polynomial, the multiplicity- $s$ solutions $e^{r t}, \dots, t^{s - 1} e^{r t}$ , repeated complex roots, and the higher-power resonance correction. ↩
Simmons, Differential Equations with Applications and Historical Notes, §22 — Higher-Order Linear Equations; Coupled Harmonic Oscillators: two masses linked by springs reduced to a fourth-order equation and its two normal-mode frequencies. ↩