We begin by looking at two variations of Newton's Model of Cooling.
dT
-- = k(A - T)
dt
dT
-- = k(sin t - T)
dt
Both of these differential equations are based on the same physical model -- the rate at which the temperature of an object changes is proportional to the difference between the ambient temperature and the object's temperature.
In the first equation the ambient temperature is a constant A. In the second equation the ambient temperature is fluctuating. When the right hand side of a differential equation
dy
-- = f(t, y)
dt
does not involve the independent variable (often time, t) then we say it is autonomous. When it does involve the independent variable we say it is nonautonomous or sometimes time-dependent. The first equation above is autonomous because the ambient temperature is constant. The second equation above is time-dependent because the ambient temperature is changing.
In both equations the constant k is positive and is determined by the physical characteristics of the object. If the object is small and poorly insulated then it responds rapidly to the ambient temperature and k is large. If the object is large and well-insulated then it responds more slowly to the ambient temperature and k is small.
We can gain a great deal of insight into differential equations
dy
-- = f(t, y)
dt
by visualizing what they say. At each point (t, y) the right side, f(t, y), of the differential equation tells us what the slope of the function y(t) would be if it passed through that point -- that is, how fast y would be changing for that particular value of y and t. Consider, for example, the differential equation
dy y
-- = ---
dt t
at the point (2, 3) we compute
dy y 3
-- = --- = ---
dt t 2
We indicate this on the graph below by drawing a line whose slope is 3/2 at the point (2, 3).
The picture below shows a slope field constructed by drawing similar lines at many points.
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The applet at the right can be used to examine the slope field for the
differential equation
where k is a positive constant. The x-axis runs from 0 to 4 and the y-axis runs from -2 to 2. The curve in the middle is the function At any point above this curve y' is negative and the slope is pointing downward. Click at a point above the curve to see this. At any point below the curve y' is positive and the slope is heading upward. Click at a point below the curve to observe this. At any point on this curve y' is zero and the slope is horizontal. Click carefully at a point right on the curve to observe this. You can construct a slope field like the one above for this differential equation by clicking at a series of points on the applet. You can clean the applet to start fresh by clicking anyplace along its gray border. |
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One way to think about a slope field is by thinking about currents in a lake.
The slope field shows the direction the currents are moving at each point.
When we are given an initial value, like y(1) = 1 this tells us
a starting point -- a place to drop a cork in the lake. Once the cork is
placed in the lake it is carried along by the currents. The path it follows
is often called a trajectory and is the graph of the solution to
that particular initial value problem.
The applet at the right shows the slope field for the differential equation Click anyplace on this graph to "drop the cork in the lake" and observe its trajectory. |
Before going on, answer the following questions.
dy
-- = 0.25 (y - 1)(5 - y), y(0) = 6
dt
dy
-- = 0.25 (y - 1)(5 - y), y(0) = 5
dt
dy
-- = 0.25 (y - 1)(5 - y), y(0) = 3
dt
dy
-- = 0.25 (y - 1)(5 - y), y(0) = 1
dt
dy
-- = 0.25 (y - 1)(5 - y), y(0) = 0
dt
The problems above illustrate very common phenomena for autonomous differential equations -- that is, equations of the form
dy
-- = f(y)
dt
in which the independent variable does not appear on the right hand side.
In this particular example, if the initial condition is along the line y = 1 or along the line y = 5 then the trajectory followed that line because the right hand side of the differential equation was zero and, thus, y stayed constant. Any value of y for which f is zero is called an equilibrium point or sometimes an equilibrium value because y will remain there "in equilibrium."
Notice that in this example the slopes on both sides of the equilibrium value y = 5 are pointing toward that line, leading trajectories toward that line. This kind of equilibrium is called attracting because trajectories that start out close to it are pulled toward it.
Similarly, the equilibrium y = 1 is called repelling because trajectories that start out nearby are pushed away.
Find all the equilibrium points of each of the autonomous differential equations below and classify each one -- is it attracting? repelling? or neither?
dT -- = k(A - T) dt
where the ambient temperature A is a constant and the constant k is positive.
Next we look at two more variations of Newton's Model of Cooling.
dT
-- = k [(65 + 0.1 t) - T]
dt
dT
-- = k [sin (2 Pi w t) - T]
dt
In both of these models the ambient temperature is changing. In the first example, the ambient temperature
is rising steadily. In the second the ambient temperature
is oscillating.
Use your CAS window to draw slope fields for these two differential equations and, based on these slope fields, discuss the behavior of these models with various different initial conditions. Discuss the effects of the constant k and, in the second model, of the constant w.