Lecture Notes in Differential Equations - Bruce E. Shapiro

9
We will study linear equations in greater detail in section 4.
Often we will be faced with a problem whose description requires not one,
but two, or even more, differential equations. This is analogous to an
algebra problem that requires us to solve multiple equations in multiple
unknowns. A system of differential equations is a collection of related
differential equations that have multiple unknowns. For example, the
variable y(t) might depend not only on t and y(t) but also on a second
variable z(t), that in turn depends on y(t). For example, this is a system of
differential equations of two variables y and z (with independent variable
t):
⎫
dy
dt = 3y + t2 sin z ⎪⎬
(1.47)
dz
dt = y − z ⎪ ⎭
It is because of systems that we will use the variable t rather than x for the
horizontal (time) axis in our study of single ODEs. This way we can have
a natural progression of variables x(t), y(t), z(t), . . . , in which to express
systems of equations. In fact, systems of equations can be quite difficult
to solve and often lead to chaotic solutions. We will return to a study of
systems of linear equations in a later section.