Lecture Notes in Differential Equations - Bruce E. Shapiro

374 LESSON 34. CRITICAL POINTS
variable with differential equation z = t, with differential equation z ′ = 1,
and adding the third equation to the system:
x ′ = cos y + x 2 (34.5)
y ′ = sin x + e az (34.6)
z ′ = 1 (34.7)
We will focus on the general two dimensional autonomous system
x ′ = f(x, y) x(t 0 ) = x 0
y ′ = g(x, y) y(t 0 ) = y 0
(34.8)
where f and g are differentiable functions of x and y in some open set
containing the point (x 0 , y 0 ). From the uniqueness theorem, we know that
there is precisely one solution {x(t), y(t)} to (34.8). We can plot this solution
as a curve that goes through the point (x 0 , y 0 ) and extends in either
direction for some distance. We call this curve the trajectory of the solution.
The xy-plane itself we will call the phase-plane.
We must take care to distinguish trajectories from solutions: the trajectory
is a curve in the xy plane, while the solution is a set of points that
are marked by time. Different solutions follow the same trajectory, but at
different times. The difference can be illustrated by the following analogy.
Imagine that you drive from home to school every day on certain road, say
Nordhoff Street. Every day you drive down Nordhoff from the 405 freeway
to Reseda Boulevard. On Mondays and Wednesdays, you have morning
classes and have to drive this path at 8:20 AM. On Tuesdays and Thursdays
you have evening classes and you drive the same path, moving in the
same direction, but at 3:30 PM. Then Nordhoff Street is your trajectory.
You follow two different solutions: one which puts you at the 405 at 8:20
AM and another solution that puts at the 405 at 3:30 PM. Both solutions
follow the same trajectory, but at different times.
Returning to differential equations, an autonomous system with initial conditions
x(t 0 ) = a, y(t 0 ) = b will have the same trajectory as an autonomous
system with initial conditions x(t 1 ) = a, y(t 1 ) = b, but it will be a different
solution because it will be at different points along the trajectory at
different times.
In any set where f(x, y) ≠ 0 we can form the differential equation
dy g(x, y)
=
dx f(x, y)
(34.9)