Phenomena in many disciplines are modeled by first-order
differential
equations. Some examples include
Mechanical Systems Mechanical Systems/Vibrations
Newton's second law asserts that the sum of the external forces
acting on
an object at any instant in time is equal to the
product of the object's mass
and acceleration. This has far
reaching implications for ODE's.
Consider a mass attached to a spring that is allowed to move in
a straight
line in the horizontal direction. Let x(t), v(t),
and a(t) denote the position,
velocity, and acceleration, respectively,
of the mass. We assume that x=0
is the
equilibrium position of the mass. Hooke's law asserts that
the force
exerted on the mass by the spring is F(t)=-kx(t), where k is
the spring
constant. Newton's second law states:
Mass times Acceleration = Sum of External Forces.
Using the fact that a(t)=x''(t) we obtain the equation
This is a linear second-order ode. It is accompanied by
the initial
position x(0) and the initial velocity x'(0).
Notice that according to Hooke's law the force exerted by the
spring
depends linearly on the position x(t). In many applications
the force depends
nonlinearly on x(t). One example is Duffing's
model. The differential
equation is
Here e is a positive constant. This is an example of a
nonlinear
second-order ode.
Finally, suppose that there is damping in the spring-mass system.
Damping
might be provided by a dashpot that exerts a continuous
force that is
proportional to the velocity (F(t)=-cv(t), where
c is a constant). The total
force is a sum of force due to the
spring and the damping. The differential
equation is
An RLC circuit consists of a resistor, an inductor, and capacitor
in series
with a voltage source. Let us assume that the resistance
is R, the inductance
is L, the capacitance is C, and the electromotive
force is E(t). Let I(t) denote
the current in the circuit and Q(t) denote
the charge on the capacitor. The
charge and current are related by Q'(t)=I(t).
Basic circuit analysis states that
the sum of the voltage drops across
the circuit elements equals the applied
voltage. The voltage drop across
the inductor is LI'(t). The voltage drop
across the resistor is RI(t).
The voltage drop across the capacitor is Q(t)/C.
Hence, the differential
equation is LI'(t)+RI(t)+Q(t)/C=E(t). Differentiating
this equation we
get:
This is an example of a linear second-order ode. To completely
solve
this problem the initial current I(0) and its derivative
I'(0) must be specified.
One-Dimensional Free Fall Motion
Suppose that a particle initially at height y_0 is thrown
straight upward
with velocity v_0. Let s(t) denote the
particle's height, v(t) the particle's
velocity, and a(t) the
objects acceleration at time t. If air resistance is neglected,
then by Newton's second law we have ma(t)=-mg. Using the
fact that
a(t)=s''(t) and eliminating the mass, we obtain the
equation
The initial position s(0) and initial velocity s'(0)=v(0) must
be specified.
This is linear second-order ode.
Now suppose that air resistance is proportional to velocity.
This is another example of a linear second-order ode.
[ODE Home] [1st-Order Home] [2nd-Order Home] [Laplace Transform Home] [Notation] [References]
Copyright © 1996 Department of Mathematics, Oregon State University
If you have questions or comments, don't hestitate to contact us.