Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 33
Numerical Methods
Euler’s Method
By a dynamical system we will mean a system of differential equations
of the form
y 1 ′ ⎫
= f 1 (t, y 1 , y 2 , . . . , y n )
y 2 ′ = f 2 (t, y 1 , y 2 , . . . , y n ) ⎪⎬
(33.1)
.
⎪⎭
y n ′ = f n (t, y 1 , y 2 , . . . , y n )
and accompanying initial conditions
y 1 (t 0 ) = y 10 , y 2 (t 0 ) = y 2,0 , . . . , y n (t 0 ) = y n0 (33.2)
In the simplest case we have a single differential equation and initial condition
(n=1)
y ′ = f(t, y), y(0) = y 0 (33.3)
While it is possible to define dynamical systems that cannot be expressed in
this form, e.g., they have partial derivatives or depend on function values
at earlier time points, we will confine our study to these equations. In
particular we will look as some of the techniques to solve the single equation
33.3. The generalization to higher dimensions (more equations) comes from
replacing all of the variables in our methods with vectors.
Programming languages, in general, do not contain methods to solve differential
equations, although there are large, freely available libraries that
can be used for this purpose. Analysis environments like Mathematica and
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