Differential equations are important because for many physical systems, one can, subject to suitable idealizations, formulate a differential equation that describes how the system changes in time. Understanding the solutions of the differential equation is then of paramount interest.

The simplest ODE is:

You can verify that any function

y(t)
=
Ce^{kt}

where
C
is
a
constant,
is
a
solution
to
this
ODE.
If
k
>
0,
this
ODE
thus
describes **exponential
growth** (for
example,
unrestricted
population
growth).
If
k
<
0,
the
interpretation
is **exponential
decay** (for
example,
the
decay
of
radioactive
atoms).

In many applications you will encounter the ODE:

where ω is a constant.

You can verify that any function

y(t) = A sin(ωt + ø) ,

where
A
and
ø
are
constants,
is
a
solution.
This
ODE
thus
describes
a
physical
system
whose
behaviour
is **periodic
in
time**.
We
say
that
the
system
undergoes **simple
harmonic
motion** with
amplitude
A,
frequency
ω,
and
phase
ø.

In the foundational courses on ODEs you will learn to identify whether an ODE can be solved explicitly (most cannot!), or whether it is necessary to solve it numerically using a computer.