Classifying Equations: Order, Linearity, ODE vs. PDE

There is no single method that solves every differential equation. Each technique — integrating factors, the characteristic equation, series, the Laplace transform — is built for one class of equation and useless outside it. Choosing a method therefore begins with placing an equation in its classes, and the classification is a fixed vocabulary worth learning before any solving.¹ The axes are independent: an equation is ordinary or partial, of some order, linear or nonlinear, homogeneous or not, all at once.

Ordinary versus partial

The first split is by how many independent variables the unknown function depends on.²

An ordinary equation governs the charge $Q (t)$ on a capacitor in a series circuit:

L \frac{d ^{2} Q}{d t ^{2}} + R \frac{d Q}{d t} + \frac{1}{C} Q = E (t),

with $t$ the only independent variable. By contrast the heat equation and the wave equation,

α^{2} \frac{\partial ^{2} u}{\partial x ^{2}} = \frac{\partial u}{\partial t}, a^{2} \frac{\partial ^{2} u}{\partial x ^{2}} = \frac{\partial ^{2} u}{\partial t ^{2}},

have $u$ depending on both position $x$ and time $t$ , so partial derivatives appear and both are PDEs. The theory of PDEs is substantially harder than that of ODEs and largely separate from it, though separation of variables reduces the classical PDEs back to ODEs.

Order

The circuit equation above is second-order, and both classical PDEs are second-order. The most general $n$ th-order ODE relates $t$ , the unknown $y$ , and its first $n$ derivatives through some function $F$ :

F (t, y, y^{'}, y^{''}, \dots, y^{(n)}) = 0.

It is usually assumed the equation can be solved for the highest derivative,

y^{(n)} = f (t, y, y^{'}, \dots, y^{(n - 1)}),

which removes an ambiguity: a single implicit equation can encode several of these explicit ones. The equation $(y^{'})^{2} + t y^{'} + 4 y = 0$ , quadratic in $y^{'}$ , splits by the quadratic formula into two distinct first-order equations,

y^{'} = \frac{- t + t ^{2} - 16 y}{2} and y^{'} = \frac{- t - t ^{2} - 16 y}{2} .

A related notion is degree: the power to which the highest-order derivative is raised once the equation is written as a polynomial in its derivatives. $(y^{'})^{2} + t y^{'} + 4 y = 0$ has order one and degree two. Degree is defined only when such a polynomial form exists (an equation containing $sin y^{'}$ or $e^{y^{'}}$ has no degree), and it plays a far smaller role than order, which every equation possesses.

Linear versus nonlinear

The most consequential axis is linearity.

The unknown and its derivatives appear only to the first power, never multiplied together, and never inside a nonlinear function. The coefficients may be any functions of the independent variable, because linearity restricts how $y$ enters, not how $t$ does; $t^{2} y^{''} + e^{t} y^{'} + (sin t) y = 0$ is linear.

Linearity is a rule about the left-hand operator:

y

and its derivatives enter only linearly, with

t

-dependence confined to the coefficients. Any product, power, or nonlinear function of

y

breaks it.

Three standard equations show what breaks linearity:

Products of the unknown with a derivative. In $y^{'''} + 2 e^{t} y^{''} + y y^{'} = t^{4}$ the term $y y^{'}$ multiplies the unknown by its own derivative; the equation is nonlinear.
A nonlinear function of the unknown. The undamped pendulum $\frac{d ^{2} θ}{d t ^{2}} + \frac{g}{L} sin θ = 0$ is nonlinear through $sin θ$ .
Products of unknowns in a system. The Lotka–Volterra predator–prey equations couple prey $x$ and predator $y$ through the products $x y$ , making each equation nonlinear.

Linear theory is highly developed and the reason so much of the subject is tractable; nonlinear equations have no comparably general theory and are approached qualitatively or numerically. When a nonlinear equation cannot be solved directly, a common recourse is linearization: replacing it near a point by a linear approximation. For small swings $sin θ \approx θ$ , and the pendulum becomes the linear $θ^{''} + (g / L) θ = 0$ , whose solutions are ordinary oscillations. Linearization is exact only in the limit, and phenomena confined to the nonlinear regime, such as limit cycles and chaos, disappear under it; it is only a local approximation.

Property	Linear	Nonlinear
Form of $F$ in $y, y^{'}, \dots$	linear (first powers, no products)	anything else
Superposition of solutions	holds (for the homogeneous case)	fails in general
General theory	complete and constructive	fragmentary; qualitative or numerical
Example	$t^{2} y^{''} + t y^{'} + 2 y = sin t$	$y^{''} + sin y = 0$

Homogeneous versus nonhomogeneous

Within the linear class, the forcing term $g (t)$ gives a further split.

Homogeneity matters because homogeneous linear equations obey superposition: any linear combination of solutions is again a solution. If $y_{1}$ and $y_{2}$ solve $y^{''} - 5 y^{'} + 6 y = 0$ , so does $c_{1} y_{1} + c_{2} y_{2}$ for every choice of constants — the structure that makes the general solution of a linear equation a finite-dimensional family, developed in the second-order chapter. Nonhomogeneous equations lose this: the difference of two solutions solves the homogeneous equation, so the full solution is one particular solution plus the homogeneous family, never an arbitrary combination on its own. (The word homogeneous is overloaded in this subject; it names a different property of first-order equations solved by the substitution $y = v x$ , treated under separable and exact equations.)

Systems

The classifications so far assume one unknown function. With two or more, one equation no longer suffices.

The Lotka–Volterra model is a two-equation first-order system,

\frac{d x}{d t} = a x - α x y, \frac{d y}{d t} = - c y + γ x y,

for prey $x (t)$ and predator $y (t)$ . Any higher-order equation converts to a first-order system by naming its derivatives as new variables, so first-order systems and the matrix methods they enable subsume the whole subject.

The independent axes of classification. An equation is placed on each branch at once — the circuit equation is an ODE, second-order, linear, nonhomogeneous, single-unknown.

Solutions

Classification sorts equations; the parallel question is what counts as solving one.

The interval is part of the statement: a function can solve an equation on one interval and fail on another where a derivative blows up. Verifying a candidate is mechanical — compute its derivatives and substitute.³

Substitution confirms a solution but does not find one, since there are too many functions to guess from; it is the one check always available. A less trivial case shows superposition and the count of constants together.

A solution of an $n$ th-order equation generally carries $n$ arbitrary constants, one contributed by each integration needed to undo the $n$ derivatives.

Existence, uniqueness, and determination

Writing an equation does not guarantee a solution exists. The equation $(y^{'})^{2} + 1 = 0$ has no real solution at all, and $(y^{'})^{2} + y^{2} = 0$ has only the single solution $y = 0$ , carrying no constant. Three questions organize the theory:⁴

Existence. Does a solution exist? Theorems guarantee one under mild conditions on $f$ ; a model that yields an equation with no solution signals a flaw in the formulation.
Uniqueness. Is it the only one? Fixing enough conditions should pick out a single solution — otherwise a found solution might not be the whole answer.
Determination. Can the solution actually be produced, exactly or approximately? Many equations have solutions inexpressible in elementary functions, which is why numerical and series methods exist alongside exact ones.

For first-order equations Picard's theorem answers the first two at once: if $f$ and $\partial f / \partial y$ are continuous on a rectangle, exactly one integral curve passes through each interior point.⁵ The precise hypotheses, and the equations where uniqueness fails, come with existence and uniqueness.

Initial versus boundary conditions

An $n$ th-order equation needs $n$ side conditions to fix its $n$ constants, and where those conditions are imposed determines the kind of problem.

Both problems constrain a second-order equation with two conditions. The IVP pins the value and slope at one end; the BVP pins the value at each end, and the middle is whatever the equation makes it.

An IVP for a well-behaved equation has exactly one solution; a BVP can have one, none, or infinitely many, depending on the equation and the interval, and that sensitivity is what makes boundary value problems generate eigenvalues and drive Fourier methods.

A brief lineage

The vocabulary above was assembled over three centuries, each term attached to the problems that forced it.⁶ Newton and Leibniz created the calculus the subject is written in; the Bernoullis turned mechanics problems into differential equations and solved the brachistochrone; Euler classified first-order equations and found integrating factors and constant-coefficient methods; Lagrange proved the structure of linear solution spaces and developed variation of parameters; Laplace lent his name to a transform and to the equation of potential theory; and Poincaré, facing equations no formula could solve, founded the qualitative, geometric study of nonlinear solutions.

A few landmarks. The early names built exact-solution methods; the later shift toward existence theory and qualitative geometry answered the equations those methods could not solve.

Given any differential equation, naming its type, order, linearity, and whether it is forced places it in a class, and the class names the method. The full historical account of these figures closes these notes.

Boyce & DiPrima, Elementary Differential Equations and Boundary Value Problems, §1.3 — Classification of Differential Equations: ODE versus PDE, order, systems, and the linear/nonlinear distinction, presented as the vocabulary needed to select solution methods. ↩ ↩²
Simmons, Differential Equations with Applications and Historical Notes, Ch. 1 §1 — Introduction: the definitions of ordinary and partial differential equations and of order, with Legendre's, Bessel's, Laplace's, the heat, and the wave equations as reference examples. ↩
Simmons, Ch. 1 §2 — General Remarks on Solutions: verifying $y = e^{2 x}$ , $y = e^{3 x}$ , and $y = c_{1} e^{2 x} + c_{2} e^{3 x}$ as solutions of $y^{''} - 5 y^{'} + 6 y = 0$ , and the rule that a solution carries arbitrary constants equal in number to the order. ↩ ↩²
Boyce & DiPrima, §1.3 — Some Important Questions: existence, uniqueness, and the practical determination of solutions, including equations with no solution or a single constant-free solution. ↩
Simmons, Ch. 1 §2 — Picard's theorem: continuity of $f$ and $\partial f / \partial y$ on a rectangle guarantees a unique integral curve through each interior point, giving the one-parameter family of a first-order equation. ↩
Boyce & DiPrima, §1.1–§1.3 — Historical Background, Parts I–III: from Newton, Leibniz, and the Bernoullis through Euler, Lagrange, and Laplace to the qualitative and computational methods associated with Poincaré and after. ↩