Question

Use direct substitution to verify that y(t) is a solution of the given differential equation in Exercise Group 1.1.9.15–20. Then use the initial conditions to determine the constants C or c1 and c2.

17. y′′+4y=0, y(0)=1, y′(0)=0, y(t)=c1cos2t+c2sin2t

18. y′′−5y′+4y=0,   y(0)=1 , y′(0)=0,   y(t)=c1et+c2e4t

19. y′′+4y′+13y=0, y(0)=1, y′(0)=0, y(t)=c1e−2tcos3t+c2e−3tsin3t

27. The growth of a population of rabbits with unlimited resources and space can be modeled by the exponential growth equation, dP/dt=kP.

Write a differential equation to model a population of rabbits with unlimited resources, where hunting is allowed at a constant rate α.

Write a differential equation to model a population of rabbits with unlimited resources, where hunting is allowed at a rate proportional to the population of rabbits.

Write a differential equation to model a population of rabbits with limited resources, where hunting is allowed at a constant rate α.

Write a differential equation to model a population of rabbits with limited resources, where hunting is allowed at a rate proportional to the population of rabbits.

30. Radiocarbon Dating.

Carbon 14 is a radioactive isotope of carbon, the most common isotope of carbon being carbon 12. Carbon 14 is created when cosmic ray bombardment changes nitrogen 14 to carbon 14 in the upper atmosphere. The resulting carbon 14 combines with atmospheric oxygen to form radioactive carbon dioxide, which is incorporated into plants by photosynthesis. Animals acquire carbon 14 by eating plants. When an animal or plant dies, it ceases to take on carbon 14, and the amount of isotope in the organism begins to decay into the more common carbon 12. Carbon 14 has a very long half-life, about 5730 years. That is, given a sample of carbon 14, it will take 5730 years for half of the sample to decay to carbon 12. The long half-life is what makes carbon 14 dating very useful in dating objects from antiquity.

Consider a sample of material that contains A(t) atoms of carbon 14 at time t. During each unit of time a constant fraction of the radioactive atoms will spontaneously decay into another element or a different isotope of the same element. Thus, the sample behaves like a population with a constant death rate and a zero birth rate. Make use of the model of exponential growth to construct a differential equation that models radioactive decay for carbon 14.

Solve the equation that you proposed in (a) to find an explicit formula for A(t).

The Chauvet-Pont-d'Arc Cave in the Ardèche department of southern France contains some of the best preserved cave paintings in the world. Carbon samples from torch marks and from the paintings themselves, as well as from animal bones and charcoal found on the cave floor, have been used to estimate the age of the cave paintings. If a particular sample taken from the Cauvet Cave contains 2% of the expected cabon 14, what is the approximate age of the sample?

Answer 1