Introductory Differential Equations using Sage - William Stein

Chapter 1
First order differential equations
But there is another reason for the high repute of mathematics: it is mathematics
that offers the exact natural sciences a certain measure of security which,
without mathematics, they could not attain.
- Albert Einstein
1.1 Introduction to DEs
Roughly speaking, a differential equation is an equation involving the derivatives of one
or more unknown functions. Implicit in this vague definition is the assumption that the
equation imposes a constraint on the unknown function (or functions). For example, we
would not call the well-known product rule identity of differential calculus a differential
equation.
dx
Example 1.1.1. Here is a simple example of a differential equation:
dt
= x. A solution
would be a function x(t) which is differentiable on some interval and whose derivative is
equal to itself. Can you guess any solutions?
In calculus (differential-, integral- and vector-), you’ve studied ways of analyzing functions.
You might even have been convinced that functions you meet in applications arise naturally
from physical principles. As we shall see, differential equations arise naturally from general
physical principles. In many cases, the functions you met in calculus in applications to
physics were actually solutions to a naturally-arising differential equation.
Example 1.1.2. Consider a falling body of mass m on which exactly three forces act:
• gravitation, F grav ,
• air resistance, F res ,
• an external force, F ext = f(t), where f(t) is some given function.
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