Differential Equations and Linear Algebra Math 2250

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Differential Equations and Linear Algebra Math 2250

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?

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.

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.

The Malthus Model for Population Growth growth or rate constant

The Malthus Model for Population Growth growth or rate constant

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.

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.

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.