Lecture 25: Introduction to Systems of Linear ODEs (Nicholson Section 3.5)
Sub Project: Linear Algebra Lecture Videos
Alternate Video Access via MyMedia | Video Duration: 49:38
Description: The lecture starts with a crash course on ordinary differential equations. If you don’t understand a single ODE then you’ve got no chance of understanding a system of ODEs… That said, if you’re short on time and don’t have time for the “why bother?” aspect then just go directly to 38:00.
0:50 --- Started with a simple ODE problem; modelling a tank of liquid with an inflow and an outflow.
7:15 --- What is a differential equation? What is a solution? What does it mean for a function to satisfy a differential equation?
11:20 --- How to find a differential equation that models the saltwater tank?
17:40 --- Presented the solution of the initial value problem. Answered the questions about the solution.
20:30 --- Now need to address the question about what’s the right flow rate to use. This involves going back to the original modelling equation and introducing a parameter for the flow rate and finding a solution that depends on both the parameter and on time. (Before the flow rate was just a number and the solution only depended on time.)
24:45 --- Introduced the “smell the coffee and wake up” problem. Rooms in an apartment are linked to one another by air vents. Presented the problem and the questions one has about the problem.
33:45 --- How to write the pair of ODEs that model the case with only two rooms.
38:00 --- wrote the system of two ODEs as an ODE for a vector in R^2; a 2x2 matrix A is involved. Gave a bird’s eye recipe for the approach.
39:50 --- The eigenvectors and eigenvalues of A.
40:40 --- wrote down the general solution (didn’t explain where it came from).
41:30 --- What we need to do to satisfy the initial condition on the ODE. Note: something that sometimes confuses students is what happens when there’s a zero eigenvalue. In this case, exp(0 t) = 1, and so your solution involves a vector that doesn’t change in time; it just sits there. In this case, the solution has a vector 50 [1;1] in addition to a vector that does depend on time 50 exp(-2/500 t) [1;-1]. Note that the vectors [1;1] and exp(-2/500 t)[1;-1] both being multiplied by the same number, 50, has to do with the initial data. Different initial data could lead to their being multiplied by different numbers.
43:45 --- Analyzed the solutions to make sure that they make sense --- Do you get what you expect as time goes to infinity?
45:00 --- presented the solutions using matlab. This allowed me to play with the more general problem, including varying the number of rooms, the flow rate, etc. You see that if there are enough rooms then people in the far rooms won’t smell enough coffee to be woken by it. Here are some hand-written notes on the coffee problem, as well as a matlab script you can explore with.
Supplemental Object Files: 
MAT188 Lecture 25 notes - Smell the Coffee and Wake_Up.pdf
MAT188 Lecture 25 notes - Smell the Coffee and Wake_Up.pdf
