Differential Equations
Equations involving a function and its derivatives. They describe nearly every continuous process in physics, biology, and engineering — how things move, decay, oscillate, diffuse, and spread. This chapter is where calculus becomes the language of dynamics.
ODE Basics
What a differential equation is. Order, linearity, general vs particular solutions, slope fields, and the existence-and-uniqueness story.
First-Order ODEs
The four standard techniques: separable, linear (with integrating factor), exact, and substitution for Bernoulli and homogeneous equations.
Second-Order Linear ODEs
The characteristic equation and its three regimes, mapped to over-, critically-, and under-damped oscillation. Plus undetermined coefficients and variation of parameters.
Systems of First-Order ODEs · Phase Plane
Multiple quantities changing together as a single point flowing through the phase plane. Vector fields, equilibria, eigenvalue classification, and Lotka-Volterra.
Series Solutions of ODEs
When closed-form methods fail, build the solution coefficient by coefficient as a power series. Ordinary points, recurrence, and Frobenius at singular points.
The Laplace Transform
Turning a differential equation into an algebraic one. The transform, the table, the convolution theorem, and the full transform–solve–invert workflow.
PDE Introduction
From one independent variable to several. The heat, wave, and Laplace equations, boundary conditions, and separation of variables.