| Title | Summary |
|---|---|
| First-Order: Modeling | Paul's On-Line Notes on modeling with first-order equations. The applications considered are mixing, population, and motion. The examples given here all reduce to solving linear equations. |
| First-Order: Equilibrium Solutions | Paul's On-Line Notes on applications of first-order autonomous equations. They cover population dynamics and then discuss more general autonomous equations. They discuss the stability of stationary (equilibrium) solutions i.e. as being stable, unstable, or semistable. They use direction fields when it would have been simpler to use phase-line portraits in the analysis. |
| First-Order Autonomous ODEs | Professor Mattuck (MIT video) uses graphical methods to analyze the behavior of the solutions first-order autonomous equations that arise in applications. Specifically, he treats a bank account problem and some population dynamics problems. The analysis would go faster if he used phase-line portraits directly. Instead, he uses direction fields to motivate something similar to phase-line portraits, which he calls "critical point analysis". Recall that "critical point" is another term for stationary point. |
| Khan Academy: Modeling Population with Simple Differental Equation | Khan Academy video introducing modeling of populations by a homogeneous linear equation with a constant coefficient. The solution technique is slow because the equation is integrated as a separable problem rather than recognizing the equation is linear and writing the solution directly. This includes only the simplest population model. |
| Khan Academy: Particular Solution Given Initial Conditions for Population | Khan Academy video solving the simple model presented in the previous video. The solution technique is slow because the equation is integrated as a separable problem. It finds the growth rate from data given in the problem. This treats only the simplest population model. |
| Khan Academy: Newton's Law of Cooling | Khan Academy video using the Newton law of cooling to model cooling by a nonhomogeneous linear equation with a constant coefficient. The solution technique is slow because the equation is integrated as a separable problem. |
| Khan Academy: Applying Newton's Law of Cooling to Warm Oatmeal | Khan Academy video using the Newton law of cooling to model cooling by a nonhomogeneous linear equation with a constant coefficient. The solution technique is slow because the equation is integrated as a separable problem. |
| Khan Academy: Modeling Population as an Exponential Function | Khan Academy video deriving the exponential solution to the simplest population model. The solution technique is slow because the equation is integrated as a separable problem. |
| Khan Academy: Logistic Differential Equation Intuition | Khan Academy video deriving the logistic equation, the simplest nonlinear population model. The logistic is autonomous. A graphical analysis is presented, but not the phase-line method. |
| Khan Academy: Solving the Logistic Equation - Part 1 | Khan Academy video presenting the analytic solution to the logistic equation. The solution technique is slow because the derivation of the partial fraction identity is inefficient, and because the integration technique is indirect. The whole solution takes two videos! |
| Khan Academy: Solving the Logistic Equation - Part 2 | Khan Academy video finishing the analytic solution to the logistic equation. The presentation in our text is better. |
| Khan Academy: Logistic Function Application | Khan Academy video applying the solution of the logistic model. |