---
title: "Classifying Equations: Order, Linearity, ODE vs. PDE"
module: Foundations
moduleNumber: 1
lessonNumber: 2
order: 102
summary: >
  Every solution method targets a specific class of equation, so the first
  question about any differential equation is which classes it belongs to. Four
  independent axes sort them: ordinary versus partial, order, linear versus
  nonlinear, and homogeneous versus nonhomogeneous. Systems, verification of a
  solution by substitution, and the split between initial and boundary value
  problems complete the vocabulary.
topics: [Foundations]
sources:
  - book: Boyce
    ref: "Ch. 1 §1.3 Classification of Differential Equations"
  - book: Simmons
    ref: "Ch. 1 §1 Introduction; §2 General Remarks on Solutions"
---

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.[^boyce-class] 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.[^sim-def]

> **Definition (ODE vs. PDE).** If the unknown function depends on a single
> independent variable (so only ordinary derivatives appear), the equation is
> an **ordinary differential equation** (ODE). If it depends on two or more
> variables, so partial derivatives appear, it is a **partial differential
> equation** (PDE).

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,

$$
\alpha^2\,\frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t},
\qquad
a^2\,\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2},
$$

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](/differential-equations/pdes-fourier-bvp/heat-wave-laplace-equations)
reduces the classical PDEs back to ODEs.

## Order

> **Definition (Order).** The **order** of a differential equation is the order
> of the highest derivative that appears in it.

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 $F$:

$$
F\bigl(t,\, y,\, y',\, y'',\, \ldots,\, y^{(n)}\bigr) = 0.
$$

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

$$
y^{(n)} = f\bigl(t,\, y,\, y',\, \ldots,\, y^{(n-1)}\bigr),
$$

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

$$
y' = \frac{-t + \sqrt{t^2 - 16 y}}{2}
\qquad\text{and}\qquad
y' = \frac{-t - \sqrt{t^2 - 16 y}}{2}.
$$

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.
$(y')^2 + t y' + 4y = 0$ 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.

> **Definition (Linear ODE).** The equation
> $F(t, y, y', \ldots, y^{(n)}) = 0$ is **linear** if $F$ is a linear function
> of $y, y', \ldots, y^{(n)}$. Equivalently, it can be written
> $$
> a_0(t)\,y^{(n)} + a_1(t)\,y^{(n-1)} + \cdots + a_n(t)\,y = g(t),
> $$
> where the coefficients $a_i(t)$ and the forcing term $g(t)$ depend on $t$
> alone. Any equation not of this form is **nonlinear**.

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.

$$
% caption: Linearity is a rule about the left-hand operator: $y$ and its
% derivatives enter only linearly, with $t$-dependence confined to the
% coefficients. Any product, power, or nonlinear function of $y$ breaks it.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize,
  bx/.style={draw, minimum width=54mm, minimum height=15mm, align=center}]
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\node[bx, draw=acc, text=acc] (lin) at (0,0)
  {$a_0(t)\,y'' + a_1(t)\,y' + a_2(t)\,y$};
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\node[bx, draw=warn, text=warn] (nl) at (0,-2.4)
  {$y\,y' \qquad (y')^2 \qquad \sin y \qquad e^{y}$};
\node[warn, anchor=south, font=\scriptsize] at (0,-1.55) {nonlinear: any one breaks it};
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$$

Three standard equations show what breaks linearity:

- **Products of the unknown with a derivative.** In $y''' + 2e^t y'' + y y' = t^4$
  the term $y y'$ multiplies the unknown by its own derivative; the equation is
  nonlinear.
- **A nonlinear function of the unknown.** The undamped pendulum
  $\dfrac{\d^2\theta}{\d t^2} + \dfrac{g}{L}\sin\theta = 0$ is nonlinear through
  $\sin\theta$.
- **Products of unknowns in a system.** The Lotka–Volterra predator–prey
  equations couple prey $x$ and predator $y$ through the products $xy$, 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 $\sin\theta \approx \theta$, and the
pendulum becomes the linear $\theta'' + (g/L)\theta = 0$, whose solutions are
ordinary [oscillations](/differential-equations/second-order-linear/mechanical-electrical-vibrations).
Linearization is exact only in the limit, and phenomena confined to the
nonlinear regime, such as [limit cycles and chaos](/differential-equations/nonlinear/phase-plane-autonomous-stability),
disappear under it; it is only a local approximation.

| Property | Linear | Nonlinear |
| --- | --- | --- |
| Form of $F$ in $y, y', \ldots$ | 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 | $t^2 y'' + t y' + 2y = \sin t$ | $y'' + \sin y = 0$ |

## Homogeneous versus nonhomogeneous

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

> **Definition (Homogeneous linear equation).** A linear equation
> $a_0(t)\,y^{(n)} + \cdots + a_n(t)\,y = g(t)$ is **homogeneous** if
> $g(t) \equiv 0$, and **nonhomogeneous** otherwise. The term $g(t)$ is the
> **forcing** (or inhomogeneity).

Homogeneity matters because homogeneous linear equations obey **superposition**:
any linear combination of solutions is again a solution. If $y_1$ and $y_2$
solve $y'' - 5y' + 6y = 0$, 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](/differential-equations/second-order-linear/homogeneous-constant-coefficients).
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 $y = vx$, treated under
[separable and exact equations](/differential-equations/first-order/separable-and-exact).)

## Systems

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

> **Definition (System of differential equations).** A set of differential
> equations in two or more unknown functions of the same independent variable,
> each equation typically involving several of the unknowns and their
> derivatives.

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

$$
\frac{\d x}{\d t} = a x - \alpha\,x y,
\qquad
\frac{\d y}{\d t} = -c\,y + \gamma\,x y,
$$

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](/differential-equations/systems/constant-coefficient-systems-phase-portraits)
and the matrix methods they enable subsume the whole subject.

$$
% caption: 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.
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\node[lf] (l2) at (-1.8,-3.2) {1st, 2nd,\\ up to $n$th};
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> **Worked example.** Classify each equation along the four axes.[^boyce-class]
>
> - $L\,Q'' + R\,Q' + \frac{1}{C}\,Q = E(t)$ (series circuit): ODE (only $t$),
>   second-order, linear ($Q, Q', Q''$ each appear to the first power with
>   coefficients in $t$), nonhomogeneous when $E(t) \neq 0$.
> - $\alpha^2\,\partial^2 u/\partial x^2 = \partial u/\partial t$ (heat
>   equation): PDE (depends on $x$ and $t$), second-order, linear, homogeneous.
> - $y''' + 2e^t y'' + y y' = t^4$: ODE, third-order, nonlinear through the
>   product $y y'$; homogeneity is defined only for linear equations, so it does
>   not apply.
> - $\dfrac{\d^2\theta}{\d t^2} + \dfrac{g}{L}\sin\theta = 0$ (pendulum): ODE,
>   second-order, nonlinear through $\sin\theta$.
> - $\d x/\d t = ax - \alpha xy$, $\d y/\d t = -cy + \gamma xy$ (Lotka–Volterra):
>   a system of two first-order ODEs, nonlinear through the products $xy$.

## Solutions

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

> **Definition (Solution).** A function $\varphi$ is a **solution** of
> $y^{(n)} = f(t, y, y', \ldots, y^{(n-1)})$ on an interval $\alpha < t < \beta$
> if $\varphi$ has derivatives through order $n$ there and substituting
> $y = \varphi(t)$ satisfies the equation identically on the interval.

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.[^sim-verify]

> **Worked example.** Verify that $y = \cos t$ solves $y'' + y = 0$.
>
> Differentiate twice: $y' = -\sin t$ and $y'' = -\cos t$. Substituting,
>
> $$
> y'' + y = -\cos t + \cos t = 0
> $$
>
> for all $t$, so $y = \cos t$ is a solution on $(-\infty, \infty)$.

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.

> **Worked example.** Show that $y = e^{2x}$ and $y = e^{3x}$ both solve
> $y'' - 5y' + 6y = 0$, and that their combination does too.[^sim-verify]
>
> For $y = e^{2x}$, $y' = 2e^{2x}$ and $y'' = 4e^{2x}$, so
>
> $$
> y'' - 5y' + 6y = (4 - 10 + 6)\,e^{2x} = 0.
> $$
>
> For $y = e^{3x}$, $y' = 3e^{3x}$ and $y'' = 9e^{3x}$, so
>
> $$
> y'' - 5y' + 6y = (9 - 15 + 6)\,e^{3x} = 0.
> $$
>
> The equation is linear and homogeneous, so the combination
> $y = c_1 e^{2x} + c_2 e^{3x}$ is again a solution. Its derivatives are
> $y' = 2c_1 e^{2x} + 3c_2 e^{3x}$ and $y'' = 4c_1 e^{2x} + 9c_2 e^{3x}$, and
> grouping by exponential,
>
> $$
> y'' - 5y' + 6y = c_1(4 - 10 + 6)e^{2x} + c_2(9 - 15 + 6)e^{3x} = 0.
> $$
>
> It carries two arbitrary constants, matching the order.

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:[^boyce-questions]

- **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.[^sim-picard] The precise hypotheses,
and the equations where uniqueness fails, come with
[existence and uniqueness](/differential-equations/first-order/existence-uniqueness-euler).

## 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.

> **Definition (IVP vs. BVP).** An **initial value problem** specifies the
> unknown and its derivatives at a single point, $y(t_0), y'(t_0), \ldots$; a
> **boundary value problem** specifies conditions at two or more distinct
> points, typically the ends of an interval.

$$
% caption: Both problems constrain a second-order equation with two conditions.
% The IVP pins the value and slope at one end; the BVP pins the value at each
% end, and the middle is whatever the equation makes it.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
\definecolor{acc}{HTML}{2A7DB0}
\definecolor{warn}{HTML}{B0562A}
% --- IVP panel ---
\begin{scope}
  \draw[->, black] (-0.1,0) -- (4.4,0) node[below, font=\scriptsize] {$t$};
  \draw[->, black] (0,-0.1) -- (0,3.1) node[left, font=\scriptsize] {$y$};
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% --- BVP panel ---
\begin{scope}[xshift=6.2cm]
  \draw[->, black] (-0.1,0) -- (4.4,0) node[below, font=\scriptsize] {$t$};
  \draw[->, black] (0,-0.1) -- (0,3.1) node[left, font=\scriptsize] {$y$};
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\end{tikzpicture}
$$

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](/differential-equations/pdes-fourier-bvp/fourier-series).

## A brief lineage

The vocabulary above was assembled over three centuries, each term attached to
the problems that forced it.[^boyce-history] 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](/differential-equations/nonlinear/phase-plane-autonomous-stability)
solutions.

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

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](/differential-equations/history-variations/historical-notes)
of these figures closes these notes.

[^boyce-class]: **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.
[^sim-def]: **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.
[^sim-verify]: **Simmons**, Ch. 1 §2 — General Remarks on Solutions: verifying $y = e^{2x}$, $y = e^{3x}$, and $y = c_1 e^{2x} + c_2 e^{3x}$ as solutions of $y'' - 5y' + 6y = 0$, and the rule that a solution carries arbitrary constants equal in number to the order.
[^boyce-questions]: **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.
[^sim-picard]: **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-history]: **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.