Differential Equations

A boundary value problem (BVP) is a differential equation paired with conditions specified at the edges (boundary) of the domain, rather than all at a single starting point as in an initial value problem. For a second-order ODE on [a,b], typical boundary conditions fix y(a) and y(b) (Dirichlet), y′(a) and y′(b) (Neumann), or a mix. Unlike initial value problems, which always have a unique solution under mild hypotheses, BVPs may have no solution, a unique solution, or infinitely many, depending on how the boundary conditions interact with the equation — this is the basis of Sturm–Liouville theory and eigenvalue problems. BVPs are the natural setting for steady-state and spatial problems: a hanging cable's shape, a beam's deflection, or the steady temperature along a rod all satisfy boundary, not initial, conditions.

The heat equation, uₜ = αu_xx, is the canonical parabolic PDE and models diffusion: how temperature, concentration, or any diffusing quantity spreads out and smooths over time. Unlike the wave equation, disturbances in the heat equation do not travel at a finite speed or preserve their shape — instead sharp features immediately blur, and (in an insulated finite domain) the solution decays toward a spatially uniform steady state. The standard solution technique on a finite rod with fixed-temperature ends is separation of variables: assume u(x,t)=X(x)T(t), which splits the PDE into two ODEs whose product solutions, weighted by a Fourier sine series matched to the initial condition, give the full solution. The exponential decay rate of each Fourier mode is proportional to the square of its mode number, so high-frequency (sharp) features die out fastest — exactly the smoothing behavior diffusion is known for.

The wave equation, uₜₜ = c²u_xx, is the canonical hyperbolic PDE and describes phenomena where disturbances propagate at a finite speed c without dissipating — vibrating strings, sound waves, electromagnetic waves, and (in higher dimensions) light and seismic waves. Unlike the heat equation, solutions of the wave equation do not smooth out or decay: an initial disturbance splits and travels outward, preserving its shape (in one dimension) as it moves. The general solution on an infinite domain is d'Alembert's formula, a sum of two traveling waves moving in opposite directions; on a finite domain with boundary conditions, solutions instead decompose into a Fourier series of standing-wave normal modes with frequencies fixed by the boundary conditions.

An equilibrium (or fixed point) of a system x′=f(x) is a state x* where f(x*)=0, so a trajectory starting exactly there stays there forever. Stability asks what happens to trajectories that start near, but not exactly at, x*: they may return (stable), stay nearby without necessarily converging (neutrally stable), or move away (unstable). For a system linearized near an equilibrium, x′=Ax with A the Jacobian at x*, the eigenvalues of A determine stability — negative real parts mean decay toward the equilibrium, positive real parts mean growth away from it. This eigenvalue criterion (the linearization or Hartman–Grobman theorem, informally) is the standard first tool for classifying equilibria of nonlinear systems.

Phase plane analysis studies a two-variable system x′=f(x,y), y′=g(x,y) by plotting trajectories in the (x,y) plane rather than as functions of time. Each point in the plane has an associated velocity vector (f,g), so the phase plane is filled with a vector field whose flow lines are the system's solution curves. Qualitative features — spirals, closed loops, straight-line convergence — reveal the long-term dynamics without needing an explicit formula for x(t) and y(t). This graphical/qualitative approach is indispensable for nonlinear systems, which rarely have closed-form solutions but whose phase portraits can still be classified near equilibrium points.

A system of first-order linear differential equations describes several interacting quantities simultaneously, x′ = Ax, where x(t) is a vector of unknown functions and A is a constant coefficient matrix. Any higher-order single ODE can be rewritten as an equivalent first-order system by introducing derivatives as new variables, so systems are the general framework underlying all ODE theory. When A has a full set of eigenvectors, the solution is a combination of exponential modes eᵗλv, one per eigenvalue-eigenvector pair, exactly mirroring the characteristic-root approach for scalar equations. Systems naturally model coupled phenomena such as predator-prey populations, multi-compartment chemical reactions, and multi-mass spring networks.

Bifurcation theory studies how the qualitative structure of a dynamical system's equilibria — how many there are, and whether each is stable or unstable — changes as a parameter is varied. A bifurcation is a value of the parameter at which this qualitative picture changes abruptly: equilibria can appear, disappear, merge, split, or swap stability. Because such transitions correspond to a linearization eigenvalue crossing zero (or the imaginary axis, for oscillatory bifurcations), bifurcation analysis builds directly on the eigenvalue-based stability criteria from equilibrium and stability theory, but tracks how those eigenvalues move as a parameter r changes rather than fixing r once and for all.

A first-order equation written as M(x,y)dx + N(x,y)dy = 0 is called exact if the left side is the total differential of some function F(x,y), i.e. dF = M dx + N dy. When this holds, the solution is simply the implicit curve F(x,y) = C, so no further integration technique is needed once F is found. The test for exactness is the mixed-partials condition ∂M/∂y = ∂N/∂x, which follows from equality of mixed partial derivatives of F. Exact equations generalize separable equations and are the starting point for the method of integrating factors when a given equation is not already exact.

An integrating factor is a function μ(x,y) that, when multiplied through a non-exact first-order equation M dx + N dy = 0, makes it exact — so it can then be solved by the exact-equation method. The most common use is on linear first-order equations y′ + P(x)y = Q(x), where μ(x) = e^{∫P dx} converts the left side into the derivative of a single product, (μy)′. Finding a general integrating factor for an arbitrary non-exact equation can be hard, but for the linear case the formula is explicit and always works. This technique is one of the two standard workhorses (alongside separation of variables) for solving first-order linear ODEs in closed form.

A linear ODE is homogeneous when its forcing term is zero: aₙ(x)y⁽ⁿ⁾ + ⋯ + a₁(x)y′ + a₀(x)y = 0. The solutions of an n-th order homogeneous linear ODE form an n-dimensional vector space, so any solution is a linear combination of n linearly independent 'basis' solutions — a fact that follows directly from linearity of the differential operator. For constant-coefficient equations, the basis solutions come from the roots of the characteristic polynomial, exactly as in the second-order case but generalized to any order. Linear independence of candidate solutions is checked with the Wronskian, a determinant that is nonzero exactly when the solutions are independent.

Variation of parameters is a general method for finding a particular solution yₚ of a non-homogeneous linear ODE ay″+by′+cy = g(x), once the homogeneous solutions y₁, y₂ are known. Unlike the method of undetermined coefficients, it works for any forcing function g(x) — not just polynomials, exponentials, sines, and cosines — because it does not guess a form but instead solves directly for coefficient functions u₁(x), u₂(x) that replace the constants C₁, C₂ in the homogeneous solution. The price is that it requires computing the Wronskian and two integrals, which can be messy but always exist in principle. It is the ODE analogue of the variation-of-constants technique used for first-order linear equations.

A first-order differential equation relates a function to its own first derivative: dy/dx = f(x,y). Rather than solving for a number, you're solving for an entire function whose rate of change matches a given rule at every point. These equations model anything that changes based on its current state — population growth, radioactive decay, cooling objects, and circuit currents.

A linear differential equation is one that is linear in the unknown function y and its derivatives — it can be written a₀(x)y + a₁(x)y′ + a₂(x)y″ + ⋯ + aₙ(x)y⁽ⁿ⁾ = b(x), where the coefficients a₀,…,aₙ and the forcing term b(x) are arbitrary functions of x (they need not themselves be linear). A second-order differential equation is the n=2 case, a₂(x)y″ + a₁(x)y′ + a₀(x)y = b(x), and is the most common order encountered in physics because Newton's second law relates acceleration (a second derivative) to force.

Separation of variables is a method for solving ordinary and partial differential equations in which algebra allows the equation to be rewritten so that each of two variables occurs on a different side of the equation. For a first-order ODE dy/dx = g(x)h(y), this means moving every y-dependent piece (including dy) to one side and every x-dependent piece (including dx) to the other, so that each side can be integrated independently, one variable at a time.

Many linear differential equations that arise in practice — Legendre's equation, Bessel's equation, Hermite's equation — have variable (non-constant) coefficients, so the characteristic-equation trick used for constant-coefficient ODEs does not apply. The power series method instead looks for a solution directly as a series y = Σₙ aₙxⁿ around a point x₀. Substituting the series into the ODE and collecting like powers of x turns the differential equation into a recurrence relation among the coefficients aₙ, which can be solved term by term. Where the series converges (guaranteed near any 'ordinary point' where the coefficients are analytic, by a theorem essentially due to Cauchy and later made rigorous by Fuchs), it defines a genuine solution, and often the series can be recognized as the expansion of a familiar function.

The method of undetermined coefficients solves nonhomogeneous linear ODEs with constant coefficients, ay″ + by′ + cy = g(x), when the forcing term g(x) is one of a small family of 'nice' functions: polynomials, exponentials, sines/cosines, or products of these. The idea is to guess a particular solution yₚ with the same functional form as g(x) but with unknown (undetermined) coefficients, substitute it into the ODE, and solve the resulting algebraic system for those coefficients. Combined with the homogeneous solution y_h found from the characteristic equation, the general solution is y = y_h + yₚ. The method fails when g(x) is not of this restricted form (in which case variation of parameters is used instead), and it requires a modification — multiplying the guess by x — whenever the guessed form already solves the homogeneous equation (resonance).

Before solving an initial value problem y′ = f(x,y), y(x₀) = y₀, it is worth asking whether a solution exists at all, and if so whether it is the only one. The Picard–Lindelöf theorem (also called the Picard existence theorem) answers both questions: if f(x,y) is continuous and Lipschitz continuous in y near (x₀,y₀), then a unique solution exists on some (possibly small) interval around x₀. Continuity alone (Peano's existence theorem) guarantees existence but not uniqueness, and the interval of existence can be much shorter than the interval where f is defined — solutions can 'blow up' in finite time even when f itself is perfectly smooth everywhere.

The Laplace transform, named after Pierre-Simon Laplace, is an integral transform that converts a function f(t) of a real variable (typically time) into a function F(s) of a complex variable s. Lowercase letters conventionally denote the original time-domain function and the corresponding uppercase letter denotes its frequency-domain transform, e.g. f(t) and F(s). Its principal use is turning differential equations in t into algebraic equations in s, since differentiation in t corresponds to multiplication by s in the transformed equation.

A partial differential equation (PDE) is an equation for an unknown function of several variables that relates the function to its partial derivatives. Where an ordinary differential equation (ODE) describes a quantity that varies in one variable (usually time), a PDE describes a field — temperature u(x,t) along a bar, deflection u(x,y) of a plate, hydraulic head h(x,y,z) in an aquifer — that varies in space and time at once. Almost every continuum model in civil, mechanical, and environmental engineering is a PDE: heat conduction, elastic deformation, groundwater seepage, and structural vibration are all governed by them.

A difference equation (also called a discrete-time dynamical system) is the discrete analogue of a differential equation: instead of relating a function to its derivatives, it relates a sequence aₙ to its own earlier terms aₙ₋₁, aₙ₋₂, etc. A linear difference equation with constant coefficients, aₙ = c₁aₙ₋₁ + c₂aₙ₋₂ + ⋯ + cₖaₙ₋ₖ + g(n), is solved with machinery that closely mirrors constant-coefficient linear ODEs: a 'characteristic equation' whose roots determine the homogeneous solution, plus a particular solution for any forcing term g(n) found by an undetermined-coefficients-style guess. Difference equations arise naturally whenever a process is sampled or updated at discrete time steps — compound interest, population census data, and any recursively-defined sequence.

Bessel functions of the first kind Jₙ(x) and second kind Yₙ(x) are the two linearly independent solutions of Bessel's differential equation x²y″ + xy′ + (x²−ν²)y = 0. They arise naturally when solving the Laplace or Helmholtz equation in cylindrical coordinates by separation of variables, and appear throughout physics and engineering: heat flow in a cylinder, vibrations of a circular drum, electromagnetic wave propagation in waveguides, and signal processing. Jₙ is bounded at x=0 for integer n; Yₙ is unbounded at x=0.

A Green's function G(x, x') for a linear differential operator L with given boundary conditions is a function satisfying L G(x,x') = δ(x−x'), where δ is the Dirac delta. It represents the response of the system to a unit point source at x'. Once G is known, the solution to L u = f is given by the integral u(x) = ∫ G(x,x') f(x') dx', which reduces any forcing function to a superposition of point-source responses. Green's functions connect ODEs to integral equations and are fundamental in physics (electrostatics, quantum mechanics, wave propagation) and engineering.

Sturm–Liouville theory provides a unified framework for the eigenvalue problems that arise from separating variables in PDEs. A Sturm–Liouville problem consists of a self-adjoint differential operator L[y] = −(p(x)y′)′ + q(x)y on an interval [a,b] with appropriate boundary conditions, and an eigenvalue equation L[y] = λw(x)y (where w > 0 is a weight function). The theory guarantees: real eigenvalues, an infinite sequence λ₁ < λ₂ < … → +∞, orthogonal eigenfunctions with respect to the weight w, and completeness (every square-integrable function on [a,b] can be expanded in the eigenfunctions). Fourier series, Bessel series, and Legendre series are all special cases.

Boundary value problems (BVPs) for PDEs ask: find a function u satisfying a PDE on a domain Ω together with conditions on the boundary ∂Ω (Dirichlet: prescribe u; Neumann: prescribe ∂u/∂n; Robin: prescribe a linear combination). The three classical linear second-order PDEs — the heat equation, wave equation, and Laplace equation — represent parabolic, hyperbolic, and elliptic types respectively. The method of separation of variables reduces each to a Sturm–Liouville eigenvalue problem, and the solution is an eigenfunction expansion (Fourier series or its generalization).

A phase portrait is a geometric picture of the trajectories of an autonomous ODE system dx/dt = F(x) drawn in the phase plane (or phase space). Instead of tracking solutions as functions of time, we visualize the collection of all possible trajectories simultaneously. Stability theory classifies equilibrium points (where F(x*)=0) by their behavior: stable (nearby solutions stay close), asymptotically stable (nearby solutions converge), or unstable. For linear systems, eigenvalues of the Jacobian determine the type: nodes, spirals, saddles, and centers.

When a second-order linear ODE has analytic (power-series expandable) coefficients, its solutions near an ordinary point are themselves analytic and can be found by substituting a power series ansatz, differentiating term-by-term, and matching coefficients to obtain a recurrence relation. Two linearly independent solutions emerge from the two free parameters a₀ and a₁, corresponding to the two-dimensional solution space guaranteed by the existence-uniqueness theorem.