Mathematics & Science
Applications of Differential Equations
A Theoretical Introduction
Many real world problems can be thought of as the analysis of a single "compartment" into which some substance is flowing at a certain rate and out of which the same substance is flowing at some, probably different, rate. Examples of such situations that come to mind are:
Our analysis of all of these example types begins with the same basic assumption:
If we let y represent the amount of the substance present in the tank at time t, then this assumption immediately becomes:
The question then becomes one of determining the two rates on the right-hand side of the above equation. The method for doing this may be different from one example to another, and is best illustrated by doing a specific example to highlight the thinking involved.
A large tank initially holds 400 gallons of water into which 1600 pounds of a certain salt has been dissolved. An inflow pipe brings in water containing 2 pounds of the same salt per gallon at a rate of 5 gallons per minute. An outflow pipe allows the fully mixed fluid in the tank to exit at the same rate of 5 gallons per minute. Find the initial value problem that models the amount of the salt, y, in the tank at time t.
First let us note that the initial condition is obvious. We are told that there are initially 1600 pounds of salt in the tank, so y(0) = 1600.
Now we address the input rate mentioned in the model. This is also fairly easy to determine. We wish to determine the rate at which the amount of salt is entering the tank with respect to time, or to put it in terms of units, how many pounds of salt per minute are entering the tank. Well, the fluid is entering at 5 gallons per minute, and each gallon is carrying 2 pounds of salt. This quickly leads to the conclusion that salt is entering at 10 pounds per minute. (A unit analysis of the inflow also supports this conclusion.)
Finally we need to determine the rate at which salt is leaving the tank. This is the trickiest of the components to determine, since the concentration of the salt in the outflow is always changing. However, we do have a variable which keeps track of the amount of salt in the tank at any time, namely y pounds. But how much of that amount goes out of the tank each minute in the outflow? Well, since the flow rates of the volume of fluid coming into the tank and the volume leaving the tank are the same, namely 5 gallons per minute, it is pretty clear that the total fluid volume in the tank will never change. It will always remain the same—the initial 400 gallons. So the concentration of the salt in the tank at any particular time, t, is the total amount of salt in the tank, y pounds, divided by the total volume of the fluid, 400 gallons. i.e. the concentration is y/400 pounds per gallon. And this is the concentration that will be flowing out of the outflow pipe. So the outgoing liquid flows at 5 gallons per minute, carrying y/400 pounds of salt per gallon. We conclude that the product of these quantities is the rate at which the salt is leaving the tank, i.e. the output rate is 5y/400 pounds per minute, or, reducing, y/80 pounds per minute.
Summarizing, we have just found:
In conclusion, then, the initial value problem which models this example is:
This particular initial value problem quickly succumbs to the methods you have been learning in the lecture part of this course. After all, the differential equation is separable. However, since we're using Mathematica, we'll have it do the heavy lifting for us, and also take advantage of some of its other abilities to explore the nature of our example further.
OK, let's move on...
ODE Laboratories: A Sabbatical Project by Christopher A. Barker
©2017 San Joaquin Delta College, 5151 Pacific Ave., Stockton, CA 95207, USA