1. Differential Equations and Linear Algebra Math 2250

2. Dynamical Systems DE’s are used to model dynamical systems.
Dynamical systems are systems that change over time.
Examples?
Population growth
Radioactive decay
Temperature of an object
Spread of rumors or diseases
Currents and Voltages in electrical systems
Others?

3. The study of differential equations has three principal goals
To discover the differential equation that describes a specified physical situation (the equation describes how the system changes).
To find--either exactly or approximately---the appropriate solution of that equation. (The solution of the equation is a function that describes the system for “appropriate” times; e.g )
3. To interpret the solution that is found.

4. The original population growth model of Thomas Malthus provides a concrete example.
The rate of change (with respect to time) of a population with constant birth and death rates is proportional to the size of the population.

5. The Malthus Model for Population Growth
growth or rate constant

6. The Malthus Model for Population Growth
growth or rate constant

8. Newton’s Law of Cooling
The time rate of change (the rate of change with respect to time t) of the temperature T(t) of a body is proportional to the difference between T and the temperature M of the surrounding medium.

9. Torricelli’s law implies that the time rate of change of the volume V of water in a draining tank is proportional to the square root of the depth y of water in the tank.

10. EX: The increase in the number of people who have heard a particular rumor in a town with a population of 20,000 is proportional to the product of the number P who have heard the rumor and the number who have not heard the rumor at time t.