What is the difference between an ordinary and a partial differential equation?

An ordinary differential equation involves derivatives with respect to a single independent variable, while a partial differential equation involves partial derivatives with respect to several variables. ODEs typically model evolution in time alone; PDEs model phenomena that vary in both space and time.

Why are initial and boundary conditions needed?

A differential equation alone has infinitely many solutions; initial conditions (values at a starting point) or boundary conditions (values at the ends of an interval) single out the particular solution describing a given physical situation, and they determine whether the problem is well posed.

Ordinary Differential Equations

Ordinary differential equations relate an unknown function of a single variable to its derivatives, providing the basic language for modeling how quantities change over time.

Definition

An ordinary differential equation is an equation involving a function of one independent variable and one or more of its derivatives; solving it means finding the functions that satisfy the relation, often subject to initial or boundary conditions.

Scope

This area covers first-order and higher-order equations, existence and uniqueness of solutions, linear systems and the matrix exponential, stability and qualitative behavior, boundary value and eigenvalue problems of Sturm-Liouville type, and analytic and series methods of solution. It is the foundation on which dynamical systems and much of mathematical modeling are built.

Sub-topics

Core questions

When does an initial value problem have a solution, and is that solution unique?
How are linear systems solved and what governs their long-term behavior?
Is a given equilibrium or solution stable under small perturbations?
How do boundary and eigenvalue problems determine the natural modes of a system?

Key theories

Existence and uniqueness theory: Under a Lipschitz condition on the right-hand side, the Picard-Lindelof theorem guarantees a unique local solution to an initial value problem, while continuity alone (Peano's theorem) yields existence without uniqueness.
Linear theory and the matrix exponential: Solutions of a linear system with constant coefficients are generated by the matrix exponential, and the structure of the coefficient matrix's eigenvalues organizes the full solution space.
Stability theory: Linearization and Lyapunov functions classify equilibria as stable, asymptotically stable, or unstable, describing whether nearby solutions converge to, remain near, or depart from a reference state.

Clinical relevance

Ordinary differential equations are the standard modeling tool across the sciences and engineering, describing mechanical motion, electrical circuits, chemical kinetics, population dynamics, and epidemic spread, and they supply the local theory underlying dynamical systems and control.

History

Differential equations grew out of the calculus of Newton and Leibniz and the mechanics of the eighteenth century. Cauchy gave the first rigorous existence proofs in the nineteenth century, Lipschitz refined the uniqueness conditions, and Poincare and Lyapunov shifted attention from explicit formulas to the qualitative and stability theory that dominates the modern subject.

Key figures

Augustin-Louis Cauchy
Rudolf Lipschitz
Henri Poincare
Aleksandr Lyapunov
Jacques Charles Francois Sturm

Seminal works

coddington1955
hartman2002
perko2001

Frequently asked questions

What is the difference between an ordinary and a partial differential equation?: An ordinary differential equation involves derivatives with respect to a single independent variable, while a partial differential equation involves partial derivatives with respect to several variables. ODEs typically model evolution in time alone; PDEs model phenomena that vary in both space and time.
Why are initial and boundary conditions needed?: A differential equation alone has infinitely many solutions; initial conditions (values at a starting point) or boundary conditions (values at the ends of an interval) single out the particular solution describing a given physical situation, and they determine whether the problem is well posed.