Download Differential Equations Applications Practice

Differential Equations Applications (Packet 03)
Differential Equations Applications
Some examples of how to go from a written scenario to a differential equation:
Example 1: Global Studies
If we say for example that a population has a growth rate of 2%, this means that:
Rate of Growth of Population = (0.02)(Current Population)
dP
 .02P
dt
Some terminology that might be confusing.... the “relative growth rate” in the example above is 0.02. The
dP
“absolute growth rate” is
.
dt
Example 2: Economics/Finances
If we say that bank account earns interest compounded at a rate of 5% of the current balance per year:
Rate balance increasing = 5%(Current Balance)
dB
 .05B
dt
Example 4: Medicine
Ms. Orr found out last year that she is allergic to Hydrocodone and had a violently ill reaction to it. Hydrocodone
bitartrate sometime is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the
body decreases at a rate proportional to the amount left in the body. The half-life of hydrocodone in the body is
3.8 hours and the usual dosage is 10mg.
a) Write a differential equation for the quantity Q, of hydrocodone bitartrate in the body at time t, in hours, since
the drug was fully absorbed.
b) Solve the differential equation given in part (a) and find the proportionality constant “k.”
c) How much of the 10 mg dose is still in the body after 12 hours?
Example 5: Environmental Sciences
When ice forms on a lake, the water on the surface freezes first. As heat from the water travels up through the ice
and is lost to the air, more ice is formed. The question is: How thick is the layer of ice as a function of time? Since
the thickness of the ice increases with time, the thickness function is increasing. In addition, as the ices gets
thicker, it insulates better, therefore we expect the layer of ice to form more slowly as time goes on. Hence the
thickness function is increasing at a decreasing rate, so its graph is concave down. Suppose y represents the
thickness of the ice as a function of time, t. Since the thicker the ice, the longer it takes the heat to get through it;
we will assume that the rate at which ice is formed is inversely proportional to the thickness. (proportionality
constant: k)
Differential Equations Applications (Packet 03)
Detectives Orr, Knaus and McKinley find a murder victim at 9am. The temperature of the body measured 90.3oF.
One hour later, the temperature of the body is 89.0oF. The temperature of the room has been maintained at a
constant 68oF.
a) Assuming the temperature, T, of the body obeys Newton’s Law of Cooling, write a differential equations
for T.
(Rate of change of temp is directly proportional to the difference in internal & external temps.)
Rate of change of temp = ± (Temp Difference)
b) Solve the differential equation to estimate the time the murder occurred.
Example 3: Chemistry and Concentration
In the 1960’s the pollution in the Great Lakes became an issue of public concern. Let us set up a model to figure
out how long it would take for the lakes to flush themselves clean, assuming no further pollutants are being
dumped in the lake. Suppose Q is the total quantity of pollutant in a lake of volume A at time t. Suppose the
clean water is flowing in at a constant rate of r and that water flows out at the same rate. Assume that the
pollutant is evenly spread throughout the lake and that clean water coming into the lake immediately mixes with
the rest of the water. Since pollutants are being taken out of the lake but not added, Q decreases, and the water
leaving the lake becomes less polluted, so the rate at which the pollutants leave decreases.
Rate Q changes = - (Rate pollutants leave in outflow)
At time t, the concentration of the pollutants is Q/V and the water containing this concentration is leaving at rate
r.
Q
Rate pollutants leave in outflow = Rate of outflow x concentration = r 
V
dQ
Q
 r 
dt
V
And then getting the constants together...
dQ
r
  Q
dt
V
Differential Equations Applications (Packet 03)
Solve the following differential equations:
1) Find the unique solution to
dy
 2 y  4 that goes through (2,5)
dx
2) Find the unique solution to
dy
y

and y(0)=1
dt 3  t
3) Solve the differential equation:
dP
 aP  b
dt
4) Find the unique solution to the second order differential equation: 1  y2  y  0 and y(0)=0