Lecture 2.1: Separation of Variables Matthew Macauley Department of - - PowerPoint PPT Presentation

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Lecture 2.1: Separation of Variables Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations M. Macauley (Clemson) Lecture 2.1: Separation of Variables

SLIDE 1

Lecture 2.1: Separation of Variables

Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations

M. Macauley (Clemson)

Lecture 2.1: Separation of Variables Math 2080, ODEs 1 / 5

SLIDE 2

Introduction

Definition

The order of an ODE is the highest derivative that appears.

Example 1 (exponential growth)

Find all solutions to y ′ = ky.

M. Macauley (Clemson)

Lecture 2.1: Separation of Variables Math 2080, ODEs 2 / 5

SLIDE 3

Exponential decay

Example 2

Find all solutions to y ′ = −ky.

M. Macauley (Clemson)

Lecture 2.1: Separation of Variables Math 2080, ODEs 3 / 5

SLIDE 4

Exponential growth vs. decay

M. Macauley (Clemson)

Lecture 2.1: Separation of Variables Math 2080, ODEs 4 / 5

SLIDE 5

Decay to a limiting value

Example 3

Find all solutions to y ′ = k(72 − y).

M. Macauley (Clemson)

Lecture 2.1: Separation of Variables Math 2080, ODEs 5 / 5