Differential Equations
Differential equations are equations relating an unknown function to its derivatives. They model nearly every continuous process in science: growth and decay, oscillations, heat transfer, wave propagation, electric circuits, biological dynamics, and economic systems. They split into ordinary differential equations (ODEs) in one independent variable and partial differential equations (PDEs) in several.
The highest derivative that appears in the equation. The equation is called linear if it can be written as a linear combination of the unknown and its derivatives with coefficients depending only on the independent variables; otherwise non-linear.
First-order ODEs
Separable equations
If dy/dx = f(x)g(y), separate and integrate:
∫ dy/g(y) = ∫ f(x) dx
Linear first-order
dy/dx + P(x)y = Q(x). The integrating factor μ = e^(∫P dx) reduces it to (μy)' = μQ.
Exact equations
M dx + N dy = 0 is exact iff ∂M/∂y = ∂N/∂x; then there exists F with dF = M dx + N dy and the solution is F = C.
Bernoulli
dy/dx + P(x)y = Q(x)y^n. Substitute u = y^(1−n) to get a linear equation in u.
Second-order linear ODEs
The standard form is
a(x) y'' + b(x) y' + c(x) y = g(x)
The general solution is y = y_h + y_p, where y_h solves the homogeneous equation (g = 0) and y_p is any particular solution.
Constant coefficients
For ay'' + by' + cy = 0, try y = e^(rx). The characteristic equation ar² + br + c = 0 has roots r₁, r₂:
| Discriminant | Roots | Solution |
|---|---|---|
| b² − 4ac > 0 | Real distinct | y = C₁e^(r₁x) + C₂e^(r₂x) |
| b² − 4ac = 0 | Real repeated | y = (C₁ + C₂x)e^(rx) |
| b² − 4ac < 0 | Complex α ± iβ | y = e^(αx)(C₁ cos βx + C₂ sin βx) |
Particular solutions
- Undetermined coefficients: guess y_p of the same family as g(x) (polynomial, exponential, sine/cosine, product).
- Variation of parameters: works for any g(x); y_p = u₁(x)y₁ + u₂(x)y₂ with constraints leading to a 2×2 linear system in u_i'.
- A linear ODE of order n requires n independent boundary or initial conditions.
- Wronskian W(y₁, y₂) = y₁y₂' − y₂y₁'. Solutions are linearly independent iff W ≠ 0.
- The principle of superposition holds for all linear equations.
- A resonance occurs when the forcing function shares the frequency of a homogeneous solution; the response grows linearly with time.
Laplace transforms
Defined as
L = F(s) = ∫₀^∞ e^(−st) f(t) dt for Re s sufficiently large
Laplace transforms convert linear constant-coefficient ODEs into algebraic equations in s. Useful pairs:
| f(t) | F(s) |
|---|---|
| 1 | 1/s |
| t^n | n!/s^(n+1) |
| e^(at) | 1/(s−a) |
| sin ωt | ω/(s² + ω²) |
| cos ωt | s/(s² + ω²) |
| f' | sF(s) − f(0) |
| f'' | s²F(s) − sf(0) − f'(0) |
Convolution: L = F(s)G(s). Especially powerful for problems with discontinuous forcing or initial conditions.
Series solutions
Near an ordinary point x₀, solutions of a linear ODE admit Taylor expansions:
y(x) = Σ a_n (x − x₀)^n
Substituting yields a recursion for a_n. Near a regular singular point, use the Frobenius method:
y(x) = (x − x₀)^r Σ a_n (x − x₀)^n
The indicial equation gives r. This approach yields the special functions of mathematical physics — Bessel, Legendre, Hermite, Laguerre, hypergeometric.
Systems of ODEs and qualitative analysis
A linear system x' = Ax has solutions built from eigenvalues and eigenvectors of A:
- All eigenvalues with negative real parts → asymptotically stable equilibrium.
- A pair of complex eigenvalues with non-zero imaginary parts → spiral/centre.
- Any eigenvalue with positive real part → unstable.
For non-linear systems, linearisation at fixed points and the phase plane (for 2-D autonomous systems) reveal qualitative behaviour: nodes, saddles, spirals, limit cycles. Lyapunov functions prove stability beyond the linear regime.
For exam problems on linear ODEs, always (i) classify the order/linearity, (ii) write down the characteristic equation, (iii) determine the form of y_p by inspection of the right-hand side, (iv) impose initial/boundary conditions last. Memorising standard particular-solution forms saves time.
Partial differential equations
PDEs of mathematical physics divide into three canonical types based on the discriminant b² − 4ac of the second-order linear PDE:
| Type | Example | Behaviour |
|---|---|---|
| Hyperbolic | Wave eq: u_tt = c² u_xx | Finite-speed propagation, characteristics |
| Parabolic | Heat eq: u_t = α u_xx | Diffusion, smoothing |
| Elliptic | Laplace's eq: u_xx + u_yy = 0 | Equilibrium, maximum principle |
Separation of variables, Fourier series, Green's functions, and modern finite-element/finite-volume methods are standard tools. PDEs underlie quantum mechanics (Schrödinger), general relativity (Einstein), fluid mechanics (Navier–Stokes), and option pricing (Black–Scholes).
Boundary-value problems and eigenvalue problems
Many PDEs reduce by separation of variables to ODEs of the form
(p(x)y')' + (q(x) + λr(x))y = 0
— a Sturm–Liouville problem. Solutions are an orthogonal basis, generalising Fourier series. Eigenvalues λ_n correspond to the normal modes of vibrating strings, drums and quantum systems.
Beyond the linear world
Modern differential-equation theory includes chaos (Lorenz, Hénon), integrable systems (KdV, Toda), stochastic differential equations (Itô calculus for finance and physics), fractional calculus, and delay differential equations. Each shares the unifying viewpoint that the description of change — the derivative — is the central object linking models to the real world.