Ordinary Differential Equations — an interactive course with visualizations, step-by-step solutions, and real engineering applications.
How the chapters connect — click any chapter to jump to its topics
Click any available chapter to explore its topics
Discover how DEs are the language of change — from coffee cooling to satellite orbits. Includes history, real examples, and interactive demos.
Master the fundamental techniques for solving first-order ODEs, from separable equations to exact and non-exact methods.
Homogeneous and non-homogeneous equations, characteristic equations, undetermined coefficients, variation of parameters, and Euler equations.
Transform methods for solving IVPs: from calculus to algebra, transfer functions, convolution, step functions, and engineering applications.
Matrix methods, eigenvalue analysis, phase portraits, stability of equilibria, and coupled systems.
Power series solutions, Frobenius method, Euler's method, Runge-Kutta, and error analysis.
Classification of PDEs, the heat and wave equations, separation of variables, Fourier series solutions, and boundary conditions.
Five fundamental methods for solving first-order ODEs. Each topic includes theory, derivations, interactive visualizations, worked examples, and practice problems.
Equations where variables can be separated to opposite sides and integrated independently.
The integrating factor method for equations linear in the unknown function.
Nonlinear equations reducible to linear form through a clever substitution.
Recognizing and solving equations that are exact differentials of a potential function.
Finding integrating factors to convert non-exact equations into exact ones.
Equations whose right-hand side depends only on \(y/x\); solved by the substitution \(v = y/x\).
Four powerful methods for solving second-order ODEs. Covers constant and variable coefficient equations, with engineering applications in vibrations, circuits, and control systems.
The big picture: why second-order DEs matter, how \(y_c\) captures dynamics and \(y_p\) captures the input response, and how all four topics connect.
Characteristic equation method with three root cases: distinct real, repeated, and complex conjugate.
The system as a gain: how it amplifies or attenuates forcing functions, with overlap detection rules.
A general method using the Wronskian to find particular solutions for any forcing function.
Euler–Cauchy equations with indicial equation method, plus reduction of order for variable coefficients.
The engineer's power tool: convert differential equations into algebraic equations, solve in the s-domain, and transform back. Covers transfer functions, frequency response, and real-world signal handling.
The big picture: why Laplace transforms turn calculus into algebra, transfer functions as a system's DNA, and the time-domain ↔ s-domain connection.
The integral definition, complete transform table, linearity property, and first shifting theorem (s-shifting).
Partial fractions for 3 cases (distinct, repeated, complex roots), Heaviside's cover-up method, pole-zero interpretation.
The complete pipeline: transform, algebra, inverse. Transfer functions, natural vs forced response, RLC circuit applications.
Unit step functions, piecewise inputs, second shifting theorem (t-shifting), convolution theorem, and impulse response.
From coupled equations to matrix methods: learn to solve systems using eigenvalues and eigenvectors, classify equilibrium points, and visualize dynamics through phase portraits.
Why systems of DEs matter: converting higher-order equations to first-order systems, motivation from engineering and biology, and the matrix formulation \(\mathbf{x}' = A\mathbf{x}\).
Matrix operations, determinants, characteristic equation, finding eigenvalues and eigenvectors — the tools you need for solving systems.
General solutions for \(\mathbf{x}' = A\mathbf{x}\): distinct real eigenvalues, complex eigenvalues, and repeated eigenvalues with generalized eigenvectors.
Interactive visualization of coupled first-order ODEs through a physical two-tank system. See how coupling terms connect equations and how eigenvalues determine dynamics.
Classifying equilibria: nodes, saddles, spirals, centers. The trace-determinant plane, stability analysis, and interactive phase portrait exploration.
When exact methods fail, two powerful approaches remain: represent solutions as infinite series near special points, or approximate solutions numerically step by step.
The big picture: why we need series solutions and numerical methods, how the topics connect, and a roadmap for the chapter.
Finding solutions as power series around ordinary points. Recurrence relations, radius of convergence, and connections to special functions.
Extending series solutions to regular singular points. The indicial equation, three cases for the second solution, and Bessel functions.
The simplest numerical method: follow the tangent line step by step. Error analysis, improved Euler (Heun's method), and stability.
The workhorse of numerical ODE solving: four slope evaluations for fourth-order accuracy. Comparison with Euler and applications to systems.
From ODEs to PDEs: learn the fundamental techniques for solving the heat equation, wave equation, and Laplace's equation using separation of variables and Fourier series. This chapter is organised into five focused pages for clarity.
The big picture: definitions, classification of PDEs, the three classical equations (heat, wave, Laplace), and a roadmap of the chapter.
Full derivation of separation of variables with the complete three-case analysis of the separation constant $\lambda$.
Detailed case-by-case treatment: non-homogeneous Dirichlet, PDE with source term, and Neumann (where $\lambda=0$ becomes non-trivial).
Same spatial three-case analysis, but second-order temporal ODE giving oscillations. Two initial conditions — displacement and velocity.
Interactive heat & wave simulators, 6-question self-assessment quiz, summary table, and quick-reference card.