§7.4 Separable differential equations (continued)

Exercises (continued)

Summary

$\frac{dy}{dt} = \frac{-2ty}{t^2 + 1}$, $y(0) = 4$

The mass of a radioactive sample decays at a rate that is proportional to its mass. - Express this fact as a differential equation for the mass $M(t)$ using $k$ for the constant of proportionality. - If the initial mass is $M_0$, find an expression for the mass $M(t)$. - The half-life of the sample is the amount of time required for half of the mass to decay. Knowing that the half-life of Carbon-14 is 5730 years, find the value of $k$ for a sample of Carbon-14. - How long does it take for a sample of Carbon-14 to be reduced to one-quarter its original mass? - Carbon-14 naturally occurs in our environment; any living organism takes in Carbon-14 when it eats and breathes. Upon dying, however, the organism no longer takes in Carbon-14. Suppose that you find remnants of a pre-historic firepit. By analyzing the charred wood in the pit, you determine that the amount of Carbon-14 is only 30% of the amount in living trees. Estimate the age of the firepit.（This approach is the basic idea behind radiocarbon dating.）

Consider the initial value problem

• Find the solution of the initial value problem and sketch its graph. - For what values of $t$ is the solution defined? - What is the value of $y$ at the last time that the solution is defined? - By looking at the differential equation, explain why we should not expect to find solutions with the value of $y$ you noted in (c).

Suppose that a cylindrical water tank with a hole in the bottom is filled with water. The water, of course, will leak out and the height of the water will decrease. Let $h(t)$ denote the height of the water. A physical principle called Torricelli's Law implies that the height decreases at a rate proportional to the square root of the height. - Express this fact using $k$ as the constant of proportionality. - Suppose you have two tanks, one with $k=-1$ and another with $k=-10$. What physical differences would you expect to find? - 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 $k$. - Solve the initial value problem for the tank in part (c), and graph the solution you determine. - How long does it take for the water to run out of the tank? - Is the solution that you found valid for all time $t$? If so, explain how you know this. If not, explain why not.

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

. - Sketch a slope field for $P(t)$ over the range $0\leq P\leq 6$. - Identify any equilibrium solutions and determine whether they are stable or unstable. - Find the population $P(t)$ assuming that $P(0) = 1$ and sketch its graph. What happens to $P(t)$ after a very long time? - Find the population $P(t)$ assuming that $P(0) = 6$ and sketch its graph. What happens to $P(t)$ after a very long time? - Verify that the long-term behavior of your solutions agrees with what you predicted by looking at the slope field.