
The Hook's stretch potential energy is
U(t) =
k
2
Z
0
L
ju
x
(x; t)j
2
dx;
Veri fy that the tota l energy E(t)=K(t)+U(t) o f the str ing is constant, i.e .,
dE
dt
=0, and conclude t hat E(t)=E(0).
Thus, in order to solve a wave equation, we requir e the initial disturbance u(x; 0) and in itial velocity u
t
(x; 0).
1.1.2 General remarks
As we saw from the above example, ODEs and PDEs differ in some significant aspects. We can
summarize the differences a s follows.
• The solution to an ODE is a function or a set of functions of only one independent variable,
whereas for a PDE, the solution depends on two or more independent variables. For the wave
equation, the solution u depends on two variables, x and t, and represents the position o f
point x at time t. For this reason, the derivatives in a partial differential equation are partial
derivatives instead of ordinary ones.
• While ODEs are related to pointwise quantities, PDEs are mathematical models of distrib-
uted systems or continua. For this reason, PDEs are sometimes considered as an infinite set
of ODEs.
• The general solution of an ODE depends on one or more constant parameters. For example,
the solution of the harmonic oscillator x¨ +
k
m
x = 0 can be expressed as a linear combination
of two fundamental solutions fc os(!
0
t); sin(!
0
t)g, where !
0
=
k
m
q
is the natural frequency
of the single mass-spring system. The solution is given by
x(t)=c
1
cos(!
0
t)+c
2
sin(!
0
t);
where c
1
and c
2
are arbitrary constant parameters. In contrast, the solution of PDEs usually
depends on arbitrary functions. For example, the solution of the wave equation u
tt
=
k
ρ
u
x x
for constant k and ρ can be of the form u(x; t)= f(x¡ct)+ g(x+ct), where c=
K
ρ
q
, and f
and g are arbitrary smooth functions.
• The geometry of the solution to an ODE is a curve, while for a PDE it is a surface in space
or g enerally a hypersurface. For example, the graph of the solution u(x; t) of a wave equation
in the space (x; t; u) is a surface and not a curve.
• The solution of the wave equation can be interpreted from a physical perspective as a trav-
eling wave moving to the right as f(x ¡ c t) and left as g(x +c t) with velocity c =
k
ρ
q
.
In contrast, the ordinary differential equations for a c oupled mass-spring system do not
reflect such a traveling speed. For example, consider a system of three masses connected to
three springs in series, where the initial displacement of the first mass is x
1
(0)=1 and the
other two are at rest without stretch. Assuming zero initial velocity for all three masses
(x_
1
(0)= x_
2
(0) = x_
3
(0)), the initial kinetic energy of the system is zero, and the system evolves
its dynamics based on its initial potential energy U(0) =
1
2
jx
1
(0)j
2
.
The following figure shows the motion of all three masses. As we observe, the motion of m
1
affects m
3
immediately, i.e., the pro pagation speed of the disturbance in the system is infinite.
1.1 From ODEs to PDEs 5