The Computable Differential Equation Lecture ... - Bruce E. Shapiro

Chapter 1
Classifying The Problem
1.1 Defining ODEs
Definition 1.1 (Ordinary Differential Equation, ODE). Let y ∈ R n be a variable 1
that depends on t. Then we define a differential equation as any equation of the
form
F (t, y, y ′ ) = 0 (1.1)
where F is any function of the 2n + 1 variables t, y 1 , y 2 , . . . , y n , y ′ 1 , y′ 2 , . . . , y′ n+1 .
Definition 1.2. We will call any function φ(t) that satisfies
a solution of the differential equation.
F ( t, φ(t), φ ′ (t) ) = 0 (1.2)
We will use the terms “Ordinary Differential Equation” and “Differential Equation”,
as well as the abbreviations ODE and DE, interchangeably. More generally,
on can include partial derivatives in the definition, in which case one must distinguish
between “Partial” DEs (PDEs) and “Ordinary” DEs (ODEs). We will leave
the study of PDEs to another class.
Equation 1.1 is, in general, very difficult, and often, impossible, to solve, either
analytically (e.g., by finding a formula that describes y), or numerically (e.g., for
example, by using a computer to draw a picture of the graph of the solution). Often
it is possible to solve 1.1 explicitly for the derivatives:
y ′ = f(t, y) (1.3)
Many important problems can be put into this form, and solutions are known to
exists for a wide class of functions, particular as a result of theorem 1.3. The class
of problems in which equation 1.1 can be converted to the form 1.3, at least locally,
is not seriously restrictive from a practical point of view. The only requirements are
1 It may be either a scalar variable, in which case n = 1, or a vector variables, in which case n
is the number of components of the vector.
1