Glossary

The boundary value of s past which a Laplace transform's defining integral converges.

A first-order equation whose right side depends on y alone, never explicitly on x.

The average number of new infections one infected person causes in an otherwise fully susceptible population.

A solution of Bessel's equation, the ODE that arises from separating the Laplacian in polar or cylindrical coordinates.

A qualitative change in a system's equilibria or their stability as a parameter crosses a critical value.

A differential equation paired with conditions imposed at two different points, rather than one starting point.

An equilibrium surrounded by closed trajectories that neither approach nor leave it.

A two-population model where each species' growth is slowed both by its own crowding and by the other species' presence.

A function of the state that stays exactly constant along every trajectory.

An operation combining two functions by sliding, multiplying, and summing their overlap.

A system rewritten (by diagonalizing A) so each new unknown function has its own independent equation.

An idealized instantaneous unit-area kick, concentrated entirely at one instant.

Writing a function as an infinite weighted sum of a Sturm-Liouville problem's own eigenfunctions.

A BVP that has only the trivial solution except at a discrete set of special parameter values.

A point where a system's rate of change is exactly zero in every component.

The theorem guaranteeing a first-order IVP has a solution, and only one, near a well-behaved starting point.

A solution written as y = (a formula in x), isolated on one side.

Seeking a series solution of the generalized form x^r times a power series, at a regular singular point.

A matrix whose columns are n linearly independent solutions of a system.

The whole family of solutions to a differential equation, one member per value of its arbitrary constant(s).

A vector one differentiation-step removed from an ordinary eigenvector, used when a repeated eigenvalue has too few eigenvectors of its own.

A polynomial solution of Hermite's equation, the ODE behind the quantum harmonic oscillator's eigenfunctions.

An equilibrium whose Jacobian has no eigenvalue with zero real part.

A solution left as an equation relating x and y together, never solved for y.

A system's response to Dirac delta forcing, starting from rest.

The quadratic in r that a Frobenius series' lowest-order term forces, whose roots are the series' possible starting exponents.

The largest interval around the initial point where a particular solution is actually defined and differentiable.

The operation recovering f(t) from its transform F(s).

The matrix of first partial derivatives of a system's right-hand side, evaluated at a point.

An operation turning a function of t into a function of s, trading calculus in t for algebra in s.

A polynomial solution of Legendre's equation, the ODE that arises from separating the Laplacian in spherical coordinates.

An isolated closed trajectory that nearby trajectories spiral toward (or away from).

An equation where the unknown function and its derivatives combine only by addition and multiplication by functions of x.

Replacing a nonlinear system near an equilibrium with the linear system given by its Jacobian there.

The classic two-species model where predators grow by consuming prey and decay otherwise, and prey grow alone but are consumed by predators.

The particular fundamental matrix, built from the same power series as e^x, that equals the identity at t=0.

How many times a root of the characteristic polynomial repeats.

An equilibrium where every nearby trajectory approaches (or leaves) along one of two straight eigen-directions.

A first-order equation whose right side depends explicitly on x, separate from y.

A system of first-order equations, not depending explicitly on the independent variable, that isn't linear.

An equation where y or a derivative is squared, multiplied by another copy of itself, or fed through a nonlinear function.

A linear differential equation whose highest derivative has order n.

A curve where one of a system's two rate functions is zero.

The highest-order derivative appearing in the equation.

A differential equation involving a function of exactly one independent variable.

A point where both coefficient functions of a linear ODE are analytic, so a plain power-series solution works.

Eigenfunctions for different eigenvalues of a Sturm-Liouville problem integrate to zero against each other, weighted by sigma.

A differential equation involving a function of two or more independent variables, and partial derivatives.

One specific member of the general solution, singled out by a condition.

The plane whose axes are a system's two unknown functions, used to plot trajectories instead of separate graphs against t.

A formula expressing each series coefficient in terms of earlier ones, produced by matching powers of x.

A singular point mild enough that xp(x) and x^2 q(x) are still analytic there, so a Frobenius series still works.

An equilibrium attracting trajectories along one direction and repelling them along another.

Boundary conditions that each involve the solution at only one endpoint, not both at once.

The single constant both sides of a separated equation are forced to equal.

Seeking a PDE's solution as a product of one function of each variable, turning the PDE into a pair of ODEs.

Solving a boundary-value problem by guessing the missing initial data, integrating forward, and adjusting the guess until the far boundary condition is met.

A genuine solution that satisfies the equation but corresponds to no value of the general solution's constant.

A three-compartment epidemic model tracking how many people are susceptible, infected, and recovered over time.

An equilibrium every nearby trajectory winds around infinitely often while approaching or leaving.

The standard form a linear second-order eigenvalue problem is written in before its eigenfunctions can be shown orthogonal.

Adding up every separated product solution, one per eigenvalue, to build the general series solution.

Several first-order equations in several unknown functions, coupled together and solved simultaneously.

The plane whose axes are a matrix's trace and determinant, used to classify an equilibrium's type at a glance.

The path a solution traces in the phase plane, as opposed to its graph against t.

A function that is 0 before a switch time and 1 from then on.

A self-sustaining oscillator whose damping term switches sign depending on the amplitude, settling into one fixed limit cycle regardless of starting point.

A solution to a system, written as a vector of functions rather than one function.