Lecture Notes in Differential Equations - Bruce E. Shapiro

12 LESSON 2. A GEOMETRIC VIEW
We can plug in numbers to get the position of our “particle” at any time
t: At t = 0, y = e 0 = 1; at t = 0.1, y = e ( 0.1) ≈= 1.10517; at t = 0.2,
y = e 0.2 ≈ 1.2214; etc. The corresponding “trajectory” is plotted in the
figure 2.1.
Figure 2.1: Solution for example 2.1. Here the y axis gives the particle
position as a function of time (the t or horizontal axis.
1.3
1.2
0.2
(0.2, e )
y
1.1
1.
(0,1)
0.1
(0.1, e )
0.9
0.1 0. 0.1 0.2 0.3
t
Since the solution of any (solvable 2 ) initial value problem dy/dt = f(t, y),
y(t 0 ) = y 0 is given by some function y = y(t), and because any function
y = y(t) can be interpreted as a trajectory, this tells us that any initial value
problem can be interpreted geometrically in terms of a dynamical (moving
or changing) system. 3 We say “geometrically” rather than “physically”
because the dynamics may not follow the standard laws of physics (things
like F = ma) but instead follow the rules defined by a differential equation.
The geometric (or dynamic) interpretation of the initial value problem y ′ =
y, y(0) = 1 given in the last example is is described by the plot of the
trajectory (curve) of y(t) as a function of t.
2 By solvable we mean any IVP for which a solution exists according to the fundamental
existence theorem (theorem 1.5). This does not necessarily mean that we can
actually solve for (find) an equation for the solution.
3 We will use the word “dynamics” in the sense that it is meant in mathematics
and not in physics. In math a dynamical system is anything that is changing in
time, hence dynamic. This often (though not always) means that it is governed by
a differential equation. It does not have to follow the rules of Newtonian mechanics.
The term “dynamical system” is frequently bandied about in conjunction with chaotic
systems and chaos, but chaotic systems are only one type of dynamics. We will not
study chaotic systems in this class but all of the systems we study can be considered
dynamical systems.