Use direct substitution to verify that y(t) is a solution of thegiven differential equation in Exercise Group 1.1.9.15–20. Then usethe initial conditions to determine the constants C or c1 andc2.

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 unlimitedresources and space can be modeled by the exponential growthequation, dP/dt=kP.

Write a differential equation to model a population of rabbitswith unlimited resources, where hunting is allowed at a constantrate ?.

Write a differential equation to model a population of rabbitswith unlimited resources, where hunting is allowed at a rateproportional to the population of rabbits.

Write a differential equation to model a population of rabbitswith limited resources, where hunting is allowed at a constant rate?.

Write a differential equation to model a population of rabbitswith limited resources, where hunting is allowed at a rateproportional to the population of rabbits.

30. Radiocarbon Dating.

Carbon 14 is a radioactive isotope of carbon, the most commonisotope of carbon being carbon 12. Carbon 14 is created when cosmicray bombardment changes nitrogen 14 to carbon 14 in the upperatmosphere. The resulting carbon 14 combines with atmosphericoxygen to form radioactive carbon dioxide, which is incorporatedinto plants by photosynthesis. Animals acquire carbon 14 by eatingplants. When an animal or plant dies, it ceases to take on carbon14, and the amount of isotope in the organism begins to decay intothe more common carbon 12. Carbon 14 has a very long half-life,about 5730 years. That is, given a sample of carbon 14, it willtake 5730 years for half of the sample to decay to carbon 12. Thelong half-life is what makes carbon 14 dating very useful in datingobjects from antiquity.

Consider a sample of material that contains A(t) atoms of carbon14 at time t. During each unit of time a constant fraction of theradioactive atoms will spontaneously decay into another element ora different isotope of the same element. Thus, the sample behaveslike a population with a constant death rate and a zero birth rate.Make use of the model of exponential growth to construct adifferential equation that models radioactive decay for carbon14.

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

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