Differential Equations Handout A Math 1b November 11 2005 These are problems for the differential equations unit of the course some topics of which are not covered in the main textbook Some of these problems apply to other topics already covered in class or in the book they will not be assigned but feel free to do them for practice 1 For each of the differential equations below do the following Sketch the slope field in the tx plane t on the horizontal axis and x on the vertical and on the same set of axes sketch representatives of the family of solutions Check your answer using the dfield Guess the general solution to the differential equation and check your answer Find solutions corresponding to the initial conditions x 0 1 x 0 0 and x 0 1 Will solutions corresponding to different initial conditions intersect dx 2t dt dx 2x ii dt dx iii t2 dt i 2 Which of the following is a solution to the first order1 differential equation dy y 1 dt a y Cet b y Cet t c y C t2 t d y Cet 1 e y Ce t 1 3 Which of the following is a solution to the second order2 differential equation y 00 9y 0 1A 2A first order differential equation involves a first derivative but no higher derivative second order differential equation involves a second derivative but no higher derivative 1 a y e3t e 3t b y Cet t c y C t2 t d y sin 3t 6 e y 5 cos 3t 4 Solutes in the bloodstream enter cells through osmosis the diffusion of fluid through a semipermeable membrane Let C C t be the concentration of a certain solute inside a particular cell The rate at which the concentration inside the cell is changing is proportional to the difference in the concentration of the solute in the bloodstream and the concentration within the cell Suppose the concentration of a solute in the bloodstream is maintained at a constant level of L gm cubic cm Write a differential equation involving dC dt Your differential equation will involve an undetermined proportionality constant Specify the sign of that constant 5 In this problems you ll construct a model for the spread of contagious disease Let N denote the total population affected by the epidemic Make the following assumptions The size of the population N is fixed throughout the time period we are considering Everyone in the population is susceptible to the disease The disease is non fatal but is long in duration so there are no recoveries during the time period we are modeling When a person is infected that individual is sick The rate that people are becoming infected is proportional to the product of infected people and healthy people since there must be some interaction between the two in order to pass along the disease a Let I I t denote the number of infected people at time t Write a differential equation involving dI dt Your equation will involve a proportionality constant Determine its sign b Think about it Look at the differential equation you wrote in part a Is the number of infected people increasing with time decreasing with time or sometimes increasing sometimes decreasing In the long run how many people does this model imply stay healthy Now use the differential equation applet dfield to check your answer 6 In applying differential equations to monetary problems we must make a a continuous model of a discrete phenomenon We do this in the following two problems Money is deposited in a bank account with a nominal annual interest rate of 4 compounded continuously This means that the account is set up so that the instantaneous rate of increase of the balance due to interest is 4 of the amount of money in the account at that moment The initial deposit of 2000 is made at a time we designate as t 0 Let M M t be the amount of money in the bank account at time t 2 a Assume there are no additional deposits or withdrawals Write a differential equation involving dM dt and give the initial condition b Solve the differential equation in part a Use the initial condition c Suppose money is being added to the account continuously at a rate of 1 000 per year and no withdrawals are made Write a differential equation involving dM dt and give the initial condition d Solve the differential equation in c using the initial condition 7 Elmer takes out a 100 000 loan for a house He pays money back at a rate of 12 000 per year The bank charges him interest at a rate of 7 25 per year compounded continuously Make a continuous model of his economic situation Use a differential equation involving dB dt where B B t is the balance he owes the bank at time t 8 When a population has unlimited resources and is free from disease and strife the rate at which the population grows is often modeled as being proportional to the population Assume that both the bee and the mosquito populations described below behave according to this model We ll make continuous models even though population numbers are discrete In both the following scenarios you are given enough information to find the poportionality constant k In one case the information allows you to find k solely using the differential equation without requiring that you solve it In the other scenario you must actually solve the differential equation in order to find k a Let M M t be the mosquito population at time t t in weeks At t 0 there are 1000 mosquitoes Suppose that when there are 5000 mosquitoes the population is growing at a rate of 250 mosquitoes per week Write a differential equation reflecting the situation Include a value for k the proportionality constant b Let B B t be the bee population at time t t in weeks At t 0 there are 600 bees When t 10 there are 800 bees Write a differential equation reflecting the situation Include a value for k the proportionality constant 9 The population in a certain country grows at a rate proportional to the population at time t with a proportionality constant of 0 03 Due to political turmoil people are leaving the country at a constant rate of 6000 people per year Assume that there is no immigration to the country Let P P t be the population at time t where t is in years We ll make a continuous model of the population a Write a differential equation reflecting the situation 3 b Solve the differential equation for P t given the information that at time t 0 there are 3 million people in the country In other words find P t the number of people in the country at time t 10 Suppose y satisfies the differential equation dy dx y2 a Describe the behavior of y assuming each one of the following initial conditions i y 0 0 ii y 0