I. Introduction
Differential Equations (Calculus 4) is the course where you learn how to model change with an equation and then solve for the function that makes that model true. Instead of being only a function of time or space, the value can be governed by an equation that relates it to its own rate of change. This is the math behind dynamic activity such as circuits, cooling, motion with drag, population models, vibrations, and feedback systems.
II. Outline
- Intro to Differential Equations
- Order, linear vs nonlinear, autonomous vs non-autonomous
- General solutions vs particular solutions, initial value problems (IVPs)
- Slope fields and qualitative behavior
- Existence & uniqueness
- First-Order Differential Equations
- Separable equations
- Linear first-order equations + integrating factor
- Exact equations
- Homogeneous substitutions (y = vx style) and Bernoulli equations
- Logistic growth and other standard models
- Numerical methods preview (Euler / improved Euler)
- Second-Order Linear Differential Equations
- Homogeneous linear ODEs and initial conditions
- Constant-coefficient equations
- Distinct real roots, repeated roots, complex roots
- Nonhomogeneous equations
- Method of undetermined coefficients
- Variation of parameters
- Mechanical/electrical analogs: mass-spring-damper, RLC-type behavior
- Resonance, beats, transient vs steady-state response
- Laplace Transforms
- Laplace basics, inverse Laplace, and IVPs
- Step functions (Heaviside), impulses (Dirac delta)
- Convolution and transfer-function style thinking
- Systems of Differential Equations
- Writing systems in matrix form
- Phase plane / direction fields for systems
- Linear systems with eigenvalues/eigenvectors
- Stability classification (nodes, spirals, saddles)
- Coupled oscillators and simple control-style models
- Series Solutions
- Power series solutions near ordinary points
- Frobenius method near singular points
- Brief exposure to special functions
III. Free Books
- Trillia Group Math Textbooks - Differential Equations
- Scroll to the bottom. These sources are dated but still worthy.
- Gerald Teschl - Ordinary Differential Equations and Dynamical Systems
- Lectures on Analytic Diff Eqs
- MSU Graduate Level ODE Course
- Ordinary Differential Equations (Open Textbook Library)
- Paul’s Online Notes — Differential Equations
IV. Video Series
- MIT Learn Diff Eqs YouTube
- Professor Leonard — Differential Equations (YouTube Playlist)
- MIT OpenCourseWare — 18.03SC Differential Equations (Fall 2011)
- Khan Academy — Differential Equations
V. See Also
- Calculus II, Calculus III, Linear Algebra, Numerical Methods, Signals & Systems, Control Systems, University Physics
