What Is a Differential Equation?

Part 14, Chapter 14: Differential Equations

Learning objectives

Classify a differential equation as ODE vs PDE, by order, and as linear vs nonlinear
Identify whether a problem is an initial value or boundary value problem
Verify that a candidate function is a solution by direct substitution
State the existence-and-uniqueness picture (Picard-Lindelof) at a high level

Where do the equations that describe the physical world come from? Almost without exception, they come from local rules: how a quantity changes here is determined by its values and gradients here. Translating "how it changes" into derivatives gives differential equations. From Newton's second law to the Black-Scholes pricing model to the SIR epidemic model, the universe is written in this language. This section establishes the vocabulary, ODE vs PDE, order, linearity, initial vs boundary conditions, so the rest of Chapter 14 can focus on the techniques that actually solve them.

ODE vs PDE

A differential equation is an equation involving an unknown function and one or more of its derivatives. The two big classes:

An ordinary differential equation (ODE) has an unknown $y(x)$ of a single variable and uses only ordinary derivatives $y', y'', \ldots$ . Example: $y' + 2y = e^x$ .
A partial differential equation (PDE) has an unknown $u(x_1, x_2, \ldots)$ of several variables and uses partial derivatives $u_x, u_{xx}, u_{xy}, \ldots$ x,uxx,uxy,ldots. Example: the heat equation $u_t = u_{xx}$ .

The order of the equation is the highest derivative that appears. $y'' + y = 0$ is second order; $u_t = u_{xx}$ is second order in $x$ but first order in $t$ , so it is second order overall.

Linear vs nonlinear

A differential equation is linear in the unknown if every occurrence of $y$ (or $u$ ) and its derivatives appears to the first power, with coefficients that are functions of the independent variable(s) only. Otherwise it is nonlinear. The linear test is mechanical: substitute $y \to c_1 y_1 + c_2 y_2$ ; if the equation distributes cleanly, it is linear.

Examples: $y'' + 3y' + 2y = \sin x$ is linear (each term has $y$ to the first power). $y' = y^2$ is nonlinear (the $y^2$ kills additivity). $y'' + \sin(y) = 0$ (the pendulum equation) is nonlinear because of the $\sin y$ .

A linear equation is homogeneous if the right-hand side is zero. The superposition principle applies only to linear homogeneous equations: if $y_1$ and $y_2$ both solve a linear homogeneous ODE, so does $c_1 y_1 + c_2 y_2$ for any constants. This is exactly why the Fourier-series approach works for the heat and wave equations: those PDEs are linear homogeneous, and modes superpose.

Initial vs boundary conditions

A general $n$ -th order linear ODE has an $n$ -dimensional family of solutions (one free constant per order). To pin down a unique solution you need $n$ extra equations. Two flavors:

Initial conditions specify $y$ and its derivatives at one point. Example: $y(0) = 1, y'(0) = 0$ for a second-order equation. An ODE plus initial conditions is an initial value problem (IVP).

Boundary conditions specify values at two or more points. Example: $y(0) = 0, y(\pi) = 0$ . An ODE plus boundary conditions is a boundary value problem (BVP).

Initial value problems are usually well-posed: the Picard-Lindelof theorem guarantees a unique solution near the initial point whenever the right-hand side is Lipschitz in $y$ . Boundary value problems are touchier: they can have zero, one, or infinitely many solutions depending on whether the boundary data resonates with an eigenmode.

Where this shows up

SIR epidemic model: The three coupled ODEs $S' = -\beta SI$ , $I' = \beta SI - \gamma I$ , $R' = \gamma I$ powered every COVID-19 forecasting dashboard. Nonlinear (because of $SI$ ), so superposition fails, you must integrate numerically.

Chemical reaction kinetics: The rate of a chemical reaction $A + B \to C$ follows the ODE $d[C]/dt = k [A][B]$ . Nonlinear in the concentrations, but the system is linear in the rate constant $k$ , which is why you can fit $k$ from data with linear regression after a log transform.

Control theory: A PID controller for a thermostat solves the linear ODE $u(t) = K_P e(t) + K_I \int e + K_D e'(t)$ Pe(t)+KIinte+KDe′(t). The choice of $K_P, K_I, K_D$ P,KI,KD determines whether the controlled system is stable (decaying exponentials) or oscillatory (complex characteristic roots).

Quantum mechanics: The Schrodinger equation $i \hbar \psi_t = -\hbar^2/(2m) \psi_{xx} + V \psi$ Vpsi is a linear PDE for the wave function. Linearity is the reason quantum states superpose, "particle in two places at once" is a literal consequence of the equation being linear.

Pause and think: The equation $y' = y^2$ is first order. What is its general solution? And what initial conditions, if any, lead to a solution that blows up in finite time? (Hint: $y(0) = 1$ .)

Try it

Classify each of the following: (a) $y'' + 4y = \sin x$ ; (b) $u_t = u_{xx} + u u_x$ t=uxx+uux; (c) $(y')^2 + y = 0$ . State order, ODE-or-PDE, and linearity.

Verify by direct substitution that $y = 3e^{2x}$ is a solution of $y' = 2y$ .

The general solution of $y'' + 9y = 0$ is $y = A\cos(3x) + B\sin(3x)$ . Find the unique solution with $y(0) = 2, y'(0) = 3$ .

State whether each problem is an IVP or BVP: (a) $y'' + y = 0, y(0) = 1, y'(0) = 0$ ; (b) $y'' + y = 0, y(0) = 0, y(\pi) = 0$ .

The Picard-Lindelof theorem requires the right-hand side $f(t, y)$ to be Lipschitz in $y$ . Show by example that $y' = \sqrt{y}$ , $y(0) = 0$ fails uniqueness (the trivial solution $y \equiv 0$ and $y = t^2/4$ both work).

A trap to watch for

The number of arbitrary constants in the general solution of an ODE equals the order of the ODE. Students sometimes write a "general solution" with too few constants and confuse it with a particular solution. The general solution of $y'' + y = 0$ is $y = A\cos x + B\sin x$ , not $y = \sin x$ , the latter is one particular instance. The general solution is the entire two-parameter family.

What you now know

You can classify any differential equation, count the free constants in its general solution, and verify candidate solutions by substitution. You can distinguish initial value problems (one unique solution per Picard-Lindelof) from boundary value problems (potentially zero, one, or infinitely many). The next section gets concrete: classical solution methods for first-order separable, first-order linear, and second-order constant-coefficient ODEs.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 14.

Boyce, W. E., DiPrima, R. C. (2017). Elementary Differential Equations (11th ed.). Wiley, ch. 1.

Evans, L. C. (2010). Partial Differential Equations (2nd ed.). AMS, ch. 1 (introduction and classification).

Strauss, W. A. (2007). Partial Differential Equations: An Introduction (2nd ed.). Wiley, ch. 1.

Hartman, P. (2002). Ordinary Differential Equations (2nd ed.). SIAM, ch. 1 (existence and uniqueness).

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