Chapter 1: First order differential equations. They present a goal: predicting a future value of a quantity modeled by a differential equation.

- Section 1.1a. Modeling via differential equations. a: Introduce the idea of a model. Distinguish between the independent variable (time), dependent variables (dependent on time) and parameters (don’t depend on time but can be adjusted).
- Section 1.1b. Modeling via differential equations. b: Unlimited population growth. P’ = k P is the equation (exponential growth). Define first-order, ordinary differential equation, equilibrium solution, initial condition, qualitative analysis. Introduce initial-value-problem, and solution. Guess and check method of finding a solution. Particular solution vs general solution. Example comparing to United States population (annual census since 1790).
- Section 1.1c. Modeling via differential equations. c: Logistic population growth. Add a second assumption (at some level of population growth will become negative). Logistic population model, nonlinear, equilibria. They do a qualitative analysis and create approximate solutions.
- Section 1.1d. Modeling via differential equations. d: Predator prey systems. Add assumptions about fox and rabbit interactions. They generate a first order system and define the solution to a system.
- Section 1.1e. Modeling via differential equations. e: Analytic, qualitative, and numerical approaches. Here they name that there are three approaches.
- Section 1.2a. Analytic technique: separation of variables. a. What is a differential equation and what is a solution?
- Section 1.2b. Analytic technique: separation of variables. b. Initial-value problems and the general solution.
- Section 1.2c. Analytic technique: separation of variables. c. Initial-value problems and the general solution.
- Section 1.2d. Analytic technique: separation of variables. d. Separable equations
- Section 1.2e. Analytic technique: separation of variables. e. Missing solutions
- Section 1.2f. Analytic technique: separation of variables. f. Getting stuck
- Section 1.2g. Analytic technique: separation of variables. g. A savings model
- Section 1.2h. Analytic technique: separation of variables. h. A mixing problem
- Section 1.3a. Qualitative technique: slope fields. a. The geometry of dy/dt = f(t,y)
- Section 1.3b. Qualitative technique: slope fields. b. Slope fields
- Section 1.3c. Qualitative technique: slope fields. c. Important special cases
- Section 1.3d. Qualitative technique: slope fields. d. Analytic versus qualitative analysis
- Section 1.3e. Qualitative technique: slope fields. e. The mixing problem revisited
- Section 1.3f. Qualitative technique: slope fields. f. An RC circuit
- Section 1.3g. Qualitative technique: slope fields. g. Combining qualitative with quantitative results
- Section 1.4a. Numerical technique: Euler’s method. a. Stepping along the slope field
- Section 1.4b. Numerical technique: Euler’s method. b. Euler’s method
- Section 1.4c. Numerical technique: Euler’s method. c. Approximating an autonomous equation
- Section 1.4d. Numerical technique: Euler’s method. d. A non-autonomous example
- Section 1.4e. Numerical technique: Euler’s method. e. An RC circuit with periodic input
- Section 1.4f. Numerical technique: Euler’s method. f. Errors in numerical methods
- Leave existence and uniqueness (1.5), equilibria and the phase line (1.6), bifurcations (1.7), integrating factors (1.9) to a later course.
- Section 1.8a. Linear equations. a. Linear differential equations
- Section 1.8b. Linear equations. b. Linearity principles
- Section 1.8c. Linear equations. c. Solving linear equations
- Section 1.8d. Linear equations. d. Qualitative analysis
- Section 1.8e. Linear equations. e. Second guessing