First-Order Differential Equations
Stub. Directly integrable equations, slope fields, separable equations, first-order linear equations, population models, equilibria and stability, existence and uniqueness, and numerical methods.
The opening module: what a differential equation and an initial-value problem are, and the basic repertoire of solution techniques for a single first-order equation — direct integration, separation of variables, and the integrating-factor method for linear equations — alongside the qualitative picture (slope fields, equilibria and their stability) and the two questions that justify trusting a solution at all: does one exist, is it unique, and if it can’t be found in closed form, how do you approximate it numerically.
Outline (TODO — flesh out each)
- Introductory lecture — what a differential equation/IVP is
- Directly integrable equations
- Slope fields
- Separable equations
- First-order linear equations — integrating factors
- Population models — exponential and logistic growth
- Equilibria and stability
- Existence and uniqueness (Picard–Lindelöf)
- Numerical methods — Euler’s method and refinements