Matrix Theory and Differential EquationsHomework 1, due 8/31/6Question 1A yoghurt culture grows exponentially in a laboratory.• After one day there are 5 grams of yoghurt.• After six days there are 100 grams of yoghurt.• How much culture was there initially?• When will there be 200 grams of yoghurt?• What is the differential equation governing the growth of the culture?Question 2A moon lander is approaching the moon at a speed of one kilometer per second.It decelerates smoothly at a constant rate of 20 meters per second per second.• Write and solve the differential equation governing its motion.• If the lander is to land on the moon with zero speed at what height abovethe moon must the deceleration begin?Question 3Find general solutions of the following differential equations, explaining your rea-soning: also discuss the large time behavior of your solutions, both to the futureand the past.•dydt− 5y = 0.•d2udt2− 4dudt− 12u = 0.•dydt= y3.Question 4A cup of coffee is cooling in a room whose temperature is held at 20 degreesCelsius. Initially the coffee has temperature 80 degrees Celsius and is cooling ata rate of14degrees per second. Find and solve the differential equation governingthe temperature of the coffee. Also plot your solution.Question 5; for discussion only, not to be gradedWater is draining out through the vertex of a conical tank (with axis vertical andvertex at the bottom; the cone has vertex semi-angle 15 degrees). Initially thedepth of water is 20 meters and the volume decreases at a rate of 0.1 cubic metersper second. Find the differential equation governing the rate of change of thedepth y meters of the water (use Torricelli’s law:dVdt= −k√y, where V cubicmeters is the volume of fluid in the cone when the depth is y meters) and plot atypical solution.2Matrix Theory and Differential EquationsHomework 2, due 9/7/6Book problems, not to be handed inDo the following book problems to prepare for quiz one, for which these are therelevant sections:• Section 1.1: 20, 22, 28, 29, 45, 46.• Section 1.2: 8, 10, 21, 28, 30, 42.• Section 1.3: 21, 22, 23, 24, 25, 26.• Section 1.4: 3, 17, 28, 48, 52, 58.• Section 1.5: 14, 22, 32, 36, 37, 38.Work to be handed in 9/7/6Question 1Consider the differential equationdydx= x − y + 2.• Plot the slope field for the differential equation.In particular plot all places with slopes of −1, −12, 0,12and 1.• Discuss the behavior of the solutions, using your slope field.• One solution y of the equation is linear in x.Find it and plot it on your slope field.3Question 2A population P hundred deer at time t years is governed by the logistic differentialequation:dPdt=P2−P25.• Plot the slope field for the differential equation including the slopes withP = 0, 1, 2, 3, 4 and 5.• Discuss the nature of the solutions.• What is the deer population when the rate of population growth is largestand what is that maximum rate of change? Explain your answer.• Solve the equation exactly with the initial condition P = 1 and plot it onyour slope field.Question 3Solve the following differential equations, with the given initial condition and foreach determine the behavior of the solution, with a sketch:• ydydx=y2+ 1x, y =34when x =54.•dydx= 6e2x−y, y = 2 when x = 0.• xdydx+ 5y = 7x2, y = 1 when x = 2.Question 4Show that for any value of the constants A and B, the function y = A cos(2t) +B sin(2t) obeys the differential equation:d2ydt2= −4y.Find the solution that obeys the initial conditions y(0) = 3, y0(0) = 4 and discussthe behavior of the solution, with a plot.4Question 5A tank contains 1000 liters of water with 100 kilograms of salt dissolved in it.Pure water is poured into the tank at a rate of 5 liters per second, is stirred thor-oughly and the mixture is pumped out at the same rate.Write and solve a differential equation for the amount of salt in the tank at time tseconds after the pouring begins.At what time will there be only one kilogram of salt remaining in the tank?5Matrix Theory and Differential EquationsHomework 3, due 9/14/6Book problems, not to be handed in• Section 1.6: 6, 10, 19, 36, 37, 50.• Section 2.1: 2, 9, 18, 26, 30, 34.• Section 2.2: 6, 10, 12, 14, 20, 22.Work to be handed in for grading 9/14/6Question 1A tank with capacity 100 gallons initially contains 50 gallons of pure water.Brine containing 2 pounds of salt per gallon is pumped in to the tank at a rate of3 gallons per minute, is thoroughly mixed and the mixture is pumped out at a rateof 2 gallons per minute.The pumping ceases when the tank is full.How much salt is in the tank then?Question 2Tank A is initially full of 100 liters of alcohol.Pure water flows into tank A at 5 liters per minute, is thoroughly mixed andpumped out at the same rate into tank B, which initially contains 50 liters of purewater.There it is again thoroughly mixed and pumped out also at 5 liters per minute.Write and solve differential equations for the amounts of alcohol in each tank.When does the amount of alcohol in tank B peak?Question 3Consider the differential equation:dydt+ 4y = 25 sin(3t).6• Show that there is a special solution of the differential equation of the formy = A cos(3t) + B sin(3t) and find the constants A and B.• Hence find the general solution and discuss its behavior, as a function oftime.• Find and plot the solution which has the initial condition y(0) = 6.Question 4Verify the following equations of differentials are exact and solve them:• (3x2+ y2)dx + (2xy − 4y3)dy = 0.• (yx−2− x)dx + (y − x−1)dy = 0.Question 5The time rate of change of a population P of rabbits is proportional to the squareroot of P .Initially P = 64 rabbits anddPdt= 2 rabbits per week.After 50 weeks about how many rabbits will there be?Explain your