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Published byJared Harrell Modified over 6 years ago

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**Differential Equations and Linear Algebra Math 2250**

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**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?

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**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.

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**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.

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**The Malthus Model for Population Growth**

growth or rate constant

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**The Malthus Model for Population Growth**

growth or rate constant

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**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.

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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.

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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.

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