Lecture 1.1: An Introduction to Ordinary Differential Equations - PowerPoint PPT Presentation

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Lecture 1.1: An Introduction to Ordinary Differential Equations Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations M. Macauley (Clemson) Lecture 1.1:

Lecture 1.1: An Introduction to Ordinary Differential Equations Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 1 / 7
Introduction to ODEs What is a Differential Equation? It is an equation involving a function and its derivatives. Example (finance) The rate of growth of an investment is proportional to the amount of the investment. Equation: P ′ ( t ) = rP ( t ). (Often, we just write P ′ = rP .) For example, consider a mutual fund that grows at a 10% rate. Note: We assume that interest is compounded continuously , i.e., at any point in 1 time, the rate of change is 10 P ( t ). M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 2 / 7
Modeling with ODEs Big idea If the rate of change of a function f is proportional to the function itself, then f ′ = rf . Example (biology) A colony of rabbits grows at a rate proportional to its size. M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 3 / 7
Modeling with to ODEs Example (chemistry) A radioactive substance decays at a rate proportional to its size. Sample question : If there are 30 grams initially, and 20 grams after one year, what is the half-life? M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 4 / 7
Modeling with ODEs Example (physics) The temperature of a cup of coffee cools at a rate proportional to the difference: “(temp. of coffee) – (ambient temp.)”. M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 5 / 7
Exponential decay What else exhibits this “decay to a limiting value” behavior in nature (approximately)? Earth’s population. Velocity of a falling object with air resistance. M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 6 / 7
Common theme: a family of solutions Some questions from calculus: What is the antiderivative of f ( t ) = 2 t ? The velocity of a car is x ′ ( t ) = 2 t . How far from home is it after t hours? An investment takes 5 years to double. How much is it worth after 8 years? M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 7 / 7

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