Differential Equations (Overview)

What a differential equation is

An algebraic equation like x² + 5x − 6 = 0 has unknown numbers; you solve it by finding the values of x that satisfy the relation. A differential equation has unknown functions; you solve it by finding the function y(x) (or u(x, t), or X(t), …) whose derivatives obey the relation.

The simplest interesting example is

y′ = k·y.

The equation says: at every point, the slope of y is proportional to its current value. The solution is y = A·e^kx for any constant A. The equation does not pick A on its own — that requires an extra initial condition like y(0) = 7. Together they yield a unique solution y = 7·e^kx.

This is the universal pattern. A differential equation determines a family of solutions; initial or boundary conditions select a specific member of the family. The number of conditions equals the order of the equation.

Worked example — solving y′ = ky by separation

Treat dy/dx as a ratio of differentials (a heuristic that can be made rigorous):

dy/dx = k·y   ⇒   dy/y = k·dx

Integrate both sides:

∫ dy/y = ∫ k·dx   ⇒   ln|y| = k·x + C

Exponentiate:

|y| = e^{k·x + C} = e^C·e^k·x

Absorb the sign and the e^C into a single constant A (allowed because the equation is linear in y):

y(x) = A·e^k·x.

Apply y(0) = 7: A = 7. Final answer y(x) = 7·e^k·x.

This same derivation models radioactive decay (k < 0), unconstrained population growth (k > 0), continuously compounded interest, RC circuit charging, and Newton's law of cooling — different physical labels on the same equation.

ODE vs PDE vs SDE vs DDE

	ODE	PDE	SDE	DDE
Independent variables	One (e.g. time t)	Multiple (e.g. x, y, z, t)	One, plus stochastic process	One, plus delay τ
Derivative type	Ordinary d/dt	Partial ∂/∂x, ∂/∂t, …	Itô / Stratonovich differential	Ordinary, but uses y(t − τ)
Canonical example	y' = k·y, F = m·a	Heat eq u_t = α·u_xx	dS = μS·dt + σS·dW	y'(t) = a·y(t − τ)
Standard solution methods	Separation, integrating factor, undetermined coefficients, Laplace transform	Separation of variables, Fourier series, Green's functions, finite element	Itô calculus, Feynman–Kac, Monte Carlo	Method of steps, characteristic equations
Closed-form solutions	Common for linear; rare for nonlinear	Rare except for linear constant-coefficient on simple domains	Rare; geometric Brownian motion is the famous case	Rare; usually solved numerically
Existence/uniqueness	Picard–Lindelöf (Lipschitz f)	Lax–Milgram (linear elliptic), Hadamard well-posedness	Existence under Lipschitz μ, σ	Step-method existence under regularity
Where it shows up	Mechanics, kinetics, control	Heat, wave, electromagnetism, fluids	Finance, population genetics, noisy physics	Biology with maturation lag, control with sensor delay

A quick ODE classification

Within ODEs, the main solvable families form a small zoo:

• Separable. y' = f(x)·g(y). Rearrange dy/g(y) = f(x)·dx and integrate.
• Linear first-order. y' + p(x)·y = q(x). Multiply by the integrating factor μ(x) = exp(∫p), then the left side becomes (μ·y)' and you integrate.
• Exact. M(x, y)·dx + N(x, y)·dy = 0 with ∂M/∂y = ∂N/∂x. Solve by finding F such that ∂F/∂x = M, ∂F/∂y = N.
• Linear constant-coefficient (any order). a_n·y⁽ⁿ⁾ + … + a₀·y = f(x). Substitute y = e^r·x into the homogeneous part to get the characteristic polynomial; combine with a particular solution (undetermined coefficients or variation of parameters) for the inhomogeneous part.
• Bernoulli, Riccati, Euler equations. Special families with their own substitutions.

Beyond this zoo, most ODEs and almost all PDEs are studied qualitatively (phase portraits, stability, bifurcation), or numerically (Runge–Kutta, finite element).

Three canonical PDEs

Linear second-order PDEs split into three families by analogy with conic sections:

• Elliptic — Laplace's equation: ∇²u = 0. Models steady-state heat, electrostatics, ideal fluid flow. Solutions are smooth and depend on boundary values everywhere at once.
• Parabolic — heat equation: ∂u/∂t = α·∇²u. Diffuses sharp initial profiles into smooth ones; information propagates at infinite speed.
• Hyperbolic — wave equation: ∂²u/∂t² = c²·∇²u. Propagates information at finite speed c with no smoothing — initial discontinuities travel along characteristics.

These three are the testbeds for almost every PDE technique you will learn: separation of variables, Fourier series, fundamental solutions, energy methods, finite differences, finite elements.

Classical applications

• Newtonian mechanics. F = m·a is a second-order ODE; F = −k·x gives the harmonic oscillator y″ + ω²y = 0.
• Population dynamics. Logistic equation y' = r·y·(1 − y/K), Lotka–Volterra predator–prey, SIR epidemic models — all ODE systems.
• Maxwell's equations. Four PDEs whose combination yields the wave equation for electromagnetic radiation, predicting light's existence before it was measured.
• Black–Scholes. A PDE for option prices derived from a geometric-Brownian-motion SDE for the stock; foundation of modern quantitative finance.
• Navier–Stokes. Nonlinear PDEs governing fluid flow; smoothness in 3D is a Clay Millennium Prize Problem ($1M for proof or counterexample).
• Schrödinger equation. A PDE whose solutions are quantum wavefunctions; entire chemistry is in principle a corollary.

Common mistakes

• Forgetting the constant of integration. A first-order ODE has a one-parameter family of solutions; dropping C gives one specific curve, not the family.
• Mistaking dy/dx for a quotient when it is not safe. Separation of variables works for separable equations; for non-separable ones, treating dy/dx as a ratio leads to wrong answers.
• Applying ODE intuition to PDEs. PDEs almost never separate. Solutions live in function spaces of two or more variables; "constants of integration" become arbitrary functions.
• Confusing initial-value and boundary-value problems. y″ + y = 0 with y(0) = 1 and y'(0) = 0 has a unique solution (cos x). The same equation with y(0) = 0 and y(π) = 0 has infinitely many (any A·sin x). Boundary problems can fail uniqueness or existence dramatically.
• Treating SDEs with ordinary calculus. Itô's lemma adds an extra ½·σ²·∂²/∂x² term that classical calculus does not. Geometric Brownian motion has mean E[S(t)] = S(0)·e^μt, but the typical path grows like e^{(μ − σ²/2)t}.
• Ignoring stability when integrating numerically. Explicit Euler diverges on stiff equations no matter how clever your initial condition. Implicit methods or adaptive step sizes are usually necessary.