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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:

SLIDE 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

SLIDE 2

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 time, the rate of change is

1 10P(t).

M. Macauley (Clemson)

Lecture 1.1: Intro to ODEs Math 2080, ODEs 2 / 7

SLIDE 3

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

SLIDE 4

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

SLIDE 5

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)

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SLIDE 6

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)

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SLIDE 7

Common theme: a family of solutions

Some questions from calculus: What is the antiderivative of f (t) = 2t? The velocity of a car is x′(t) = 2t. 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