Plain-language explanation.
Differential equations describe how quantities change over time or space by relating a function to its derivatives. They are the primary mathematical language of physics, engineering, biology, economics, and any domain that models dynamic change.
Core concepts and standard treatment.
Core differential equations covers ordinary differential equations (ODEs — first-order: separable, linear, exact; higher-order: characteristic equations, undetermined coefficients, variation of parameters), systems of ODEs (matrix form, phase plane analysis — nullclines, equilibria, stability), Laplace transforms (solving IVPs via algebraic manipulation, convolution theorem, transfer functions), and introduction to partial differential equations (PDEs — the heat equation, wave equation, and Laplace's equation — separation of variables, Fourier series).
Deeper theory, debates and edge cases.
Advanced differential equations covers dynamical systems theory (linearisation, Lyapunov stability, bifurcation theory — saddle-node, Hopf bifurcation, limit cycles; chaos theory — Lorenz attractor, sensitive dependence on initial conditions, Lyapunov exponents), PDEs in depth (Fourier and Laplace transforms for PDEs, method of characteristics, Green's functions, distributions/generalised functions — Dirac delta), numerical methods for ODEs and PDEs (Runge-Kutta methods, finite difference, finite element, spectral methods), and stochastic differential equations (Itô calculus, Fokker-Planck equation, Langevin dynamics).
How it is applied in practice.
At the applied mathematician, engineer, and quantitative modeller level, differential equations are foundational: control systems (PID controller design, Bode plots, Nyquist stability — Laplace domain); computational fluid dynamics (Navier-Stokes equations — finite volume methods, OpenFOAM, Fluent); epidemiological modelling (SIR/SEIR models — COVID-19 projections, R₀, herd immunity threshold); financial modelling (Black-Scholes PDE, interest rate models — Vasicek, Hull-White); and neural ODEs (continuous-depth neural networks — Chen et al., adjoint method for training).