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7.4. SEPARABLE DIFFERENTIAL EQUATIONS
(c) Suppose you have a tank for which the height decreases at 20 inches per minute
when the water is filled to a depth of 100 inches. Find the value of k.
(d) Solve the initial value problem for the tank in part (c), and graph the solution
you determine.
(e) How long does it take for the water to run out of the tank?
(f) Is the solution that you found valid for all time t? If so, explain how you know
this. If not, explain why not.
4. The Gompertz equation is a model that is used to describe the growth of certain
populations. Suppose that P(t) is the population of some organism and that
dP
dt
= −P ln
P
3
= −P(ln P − ln 3).
(a) Sketch a slope field for P(t) over the range 0 ≤ P ≤ 6.
(b) Identify any equilibrium solutions and determine whether they are stable or
unstable.
(c) Find the population P(t) assuming that P(0) = 1 and sketch its graph. What
happens to P(t) after a very long time?
(d) Find the population P(t) assuming that P(0) = 6 and sketch its graph. What
happens to P(t) after a very long time?
(e) Verify that the long-term behavior of your solutions agrees with what you
predicted by looking at the slope field.