Phenomena in many disciplines are modeled by first-order
differential
equations. Some examples include
Consider a ball of mass m falling under the influence of gravity.
Let y(t)
denote the height of the ball and v(t) denote the velocity
of the ball. (In our
coordinate system positive is upward.)
According to Newton's law, mass
times acceleration equals force,
we get the following differential equations:
The first equation can be simplified to read v'=-g. These
two differential
equations can be accompanied by initial
conditions: the initial position y(0)
and velocity v(0).
The above problem can be solved easily. The equations are
both
directly integrable.
If you have taken physics class you should remember
that
where v_0 is the initial velocity and y_0 is the initial
position. Suppose we
now assume that the ball is subject to air
resistance. A simple model is
that the force of air resistance is
proportional to velocity. In this case, the
new differential
equations are
The term -kv(t) represents air resistance and k is a constant.
The minus
sign means that air resistance acts in the direction
opposite to the motion of
the ball.It is more difficult to
solve this problem exactly. So we will not give the
solution
here. This is an example of a
linear ode.
Consider a series RC (resistor and capacitor in series) circuit
with voltage
source V(t). Let I(t) denote the current. The
differential equation for the
current is
Here R is the resistance of the resistor and C is the
capacitance of the capacitor
(both are constants).
R, C and V(t) and the intial current I(0) must be specified.
This is
another example of a
linear ode.
The simplest population growth model, the Malthusian model, states that
the
rate of change of population is proportional to
the population. In
mathspeak we have
Here P(t) is the population at time t and a is a constant.
(We assume that the
population is a continuous function for
simplicity. If we assume P(t) represents
number of people,
then obviously P(t) can take only integer values. A different
type of analysis is required.)
This is an example of
a linear separable
ode. The exact solution to the problem is
If a is positive, the populations grows exponentially for all
time. This is unrealistic.
A more realistic model is the
logistic model
Here a and b are constants. In this model P'(t) is a sum of
positive and negative
terms (assuming P(t) is non-negative).
If P(t) is sufficiently large, P'(t) is negative.
It turns out as t increases
the poplulation approaches b. b is the
carrying capacity of
this model. This is an example of
a nonlinear separable
ode.
Let T(t) denote the temperature of an object and let
M(t) be the temperature
of the surrounding environment.
Newton's law of cooling states that the rate of
change of temperature
of the object is proportional to the difference
between
the object and environment temperatures. The
differential equation is
Here k is a positive constant. Notice the negative sign.
If the object is at a
higher temperature than the
environment, then T'(t) is negative and the
temperature decreases, agreeing
with intuition. M(t), k, and the initial
temperature T(0) must be
specified. This is another
example of a linear ode.
Consider a tank with volume 100L containing a salt solution.
Suppose a
solution with 2kg/L of salt flows into the tank at
a rate of 5L/min. The
solution in the tank is well-mixed.
Solution flows out of the tank at a
rate of 5L/min. If
initially there is 20kg of salt in the tank, how much salt
will be in the tan as a function of time?
This is an example of compartment problem. Let x(t) be the
amount
of salt in the tank in kg. x'(t) is equal to the
rate at which salt enters the
tank minus the rate at which
salt leaves the tank. The rate at which salt
enters the
tank is (5L/min)(2kg/L)=10kg/min. The rate at which salt
leaves
the tank is equal to the rate of flow of solution out of
the
tank times the
concentration of salt in the solution. The
concentration of salt is x(t)/100L.
Hence the rate of
outflow of salt is (5L/min)(x(t)/100L). The differential
equation for the amount of salt is
The is another example of a linear ode. The solution of the ode is
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