Download Section 1.3 Application Handout

Math 220
1.3 Differential Equations as Mathematical Models (1st Order)
1. Radioactive Decay: Suppose it is assumed that a radioactive substance decays at
a rate that is proportional to the amount of the substance (A) present at time, t.
Determine a differential equation for the given scenario.
2. Malthusian Population Model: In the late 18th century, the economist Thomas
Malthus constructed one of the earliest models for population growth. The
assumption behind the model is that the rate at which the population of a region
grows is proportional to the size of the total population at that time.
a. Determine a differential equation for the population, P(t).
b. Assume that the constant of proportionality is equal to 0.02. Using Maple,
sketch the slope field for the Malthusian Population Model. What do we
expect will happen to the population as t   ? Is this realistic? Explain.
3. Newton’s Law of Cooling/Warming: Experiments have shown that given
certain conditions, a good approximation for the temperature of an object can be
obtained by using Newton’s law of cooling/warming: The temperature of a body
changes at a rate that is proportional to the difference between the temperature of
outside medium ( Tm ) and the temperature of the body itself (T).
a. Express Newton’s Law of Cooling/Warming as a differential equation.
b. Suppose a chicken is taken out of the freezer and placed on the kitchen
table. What is a reasonable assumption for the initial temperature of the
chicken? The temperature of the surrounding medium?
c. Using the information from part b and assuming that the constant of
proportionality is equal to -.025, set up a specific differential equation for
the temperature of the chicken. Using Maple, produce a sketch of the
slope field for the DE. Have Maple sketch the particular solution based on
the stated initial condition. Explain what happens to the temperature of
the chicken as time passes.
4. Falling Bodies: According to Newton’s 2nd Law of Motion, when a net force
acting on a body is not zero, then the net force is proportional to the acceleration;
specifically F  ma . If we assume that the only force acting on the falling body
is gravity (i.e. we neglect air resistance), what is the differential equation for the
velocity of the body at time, t? What is the differential equation related to the
position of the body at time, t? (Let a positive sign represent upward direction and
negative sign represent downward direction).
5. Falling Bodies with Air Resistance: Using the model from #4, we will now take
the force of air resistance into account. Assuming that the air resistance force for
a falling body with mass, m, is proportional to the body’s instantaneous velocity
and acts in direct opposition to the force of gravity, set up a differential equation
for the object’s velocity. Also, set up a DE for the position of the body at time, t.
(For this example, let a positive sign represent downward direction and a negative
sign represent upward direction).
6. Series Circuits: According to Kirchhoff’s Second Law (for voltage) the
impressed voltage E(t) on a closed loop circuit is equal to the sum of the voltage
drops across the circuit. The voltage drops are listed below for the components
contained in a(n) RLC circuit:
Component Voltage Drop
Inductor (L)
di
L
dt
Resistor (R) iR
Capacitor
1
q
(C)
C
With i representing current and q representing charge, the relationship between them is
dq
described by i 
.
dt
a. Determine the differential equation for an RL circuit in terms of the current, i(t).
b. Determine the differential equation for an RLC circuit in terms of the charge, q(t).
7. Mixtures
Suppose a salt solution (mixture of water and salt, frequently referred to as “brine”) is
pumped into a tank already holding fluid, is well stirred and then pumped out of the tank.
If we let A = the amount of salt in the tank at time, t, then the rate of change of salt in the
tank at time, t, can be expressed as:
dA  input rate   output rate 

  Rin  Rout

dt  of salt   of salt

We usually know the concentration of the salt and the flow rate of the fluid. In general,
how would we determine the input and output rates of salt if given the concentration of
salt and the flow rate of the solution?
Consider the following scenarios:
a) A mixing tank initially holds 300 gallons of brine (salt solution). A salt solution
is pumped into the tank at a rate of 3 gal/min. The concentration of the salt in the
mixture coming into the tank is 2 lbs/gal. The solution is stirred and then pumped
out at the same rate as the solution flowing in. Determine a DE to model the
given scenario.
b) 50 gallons of brine initially containing 20 lbs of salt are in a tank into which 2
gallons of pure water run each minute with the same amount of mixture flowing
out each minute. Determine a DE for modeling the rate of change of the salt.