Differential Equations Spring 2007 Assignments

Differential Equations Spring 2007 Assignments

Homework 1, due 1/10/7 Read the first two chapters of the book up to the end of section 2.4. Prepare for the first quiz on Friday 10th January (material up to the end of section 2.2). Do the following problems to be handed in for grading. Question 1 Using dfield, or otherwise, sketch the direction field for the following logistic differential equation: dy dt = y (4 y). 8 Give the steady state solutions. Discuss the behavior of the solution for each of the following initial conditions: y(0) = 6, y(0) = 1, y(0) = 1. If a solution is bounded for all time, what can we say about the initial value of y? Explain your answer. Question 2 Using dfield, or otherwise, sketch the direction field for the following differential equation: dy dt = y t. Discuss the behavior of the general solution and find a specific (linear) solution, explaining how you see this solution from the direction field. 2

Question 3 A population of deer changes at a rate proportional to the number of deer multiplied by the difference between the number of deer and 1500. Initially the population is 300 deer. Initially the population is growing at a rate of 100 deer per year. Write the differential equation governing the population. Using dfield, or otherwise, roughly sketch the appropriate solution curve to your differential equation and discuss what happens to the population as time goes by. What is the maximum rate of change of the deer population? Explain your answer. Question 4 A pizza is removed from an oven at a temperature of 220 degrees Fahrenheit in a room held at 70 degrees Fahrenheit. Its cooling rate initially is 10 degrees per minute. Using Newton s Law of cooling, write and solve a differential equation for its cooling rate. Hence find how long it takes for the pizza to cool to a temperature of 140 degrees Fahrenheit. Question 5 A rocket approaching the planet Pluto at a speed of 1000 kilometers per second slows down by decelerating at a constant rate of 10 meters per second per second. Write and solve the differential equation governing its motion. If the rocket is to arrive at the surface of Pluto with zero velocity, at what distance from Pluto, and how long before it arrives at Pluto, should the rocket start its deceleration? 3

Homework 2, due to be discussed in class on Friday 12th January and to be handed in Wednesday 17th January 2007 Read the first three chapters of the book up to the end of section 3.4. Do the assigned book exercises at least to the end of chapter two. Do the following problems to be handed in for grading. Question 1 Using dfield, or otherwise, sketch the direction field for the following linear differential equation: dy = 2t y. dt Show that there is a linear solution to this differential equation and hence, or otherwise, find the general solution. Determine the solution that passes through the origin and sketch it on the same graph as your direction field. Question 2 Solve the following differential equation and discuss the behavior of the solution, both forward and backward in time, including a sketch of your solution: Question 3 dy dt = yt, y(2) = 1. t 2 + 5 A freely falling body hits the ground. In the last second of its motion, it is observed to travel 240 feet downwards. Assuming it was initially at rest, from what height did it fall and for how long did it fall (ignore air resistance) and how fast was it traveling when it hit the ground? 4

Question 4 Consider the differential equation: dy dt = 2y + 5 sin(t). Show that there is a solution of the form y = A cos(t) + B sin(t), for suitable constants A and B and determine A and B. Find the general solution of the differential equation. Find the solution with the initial condition y(0) = 3 and discuss its behavior as a function of time, with a sketch. Question 5 A body of mass 2 kilograms falls from rest under gravity in a liquid where the retardation force due to friction is (1.4)v Newtons, where v meters per second is its downward velocity. What is the limiting velocity of the fall? Explain your answer. How long does it take for the body to reach a speed of 90 percent of its limiting velocity? Explain your answer. How long does it take the body to fall a distance of 100 meters? Explain your answer. 5

Homework 3, to be handed in Wednesday 24th January 2007 Read the first four chapters of the book up to the end of section 4.2. Do the assigned book exercises to the end of chapter three. Prepare for the quiz Friday, topics: tanks, modeling, linear equations. Do the following problems to be handed in for grading. Question 1 A tank contains one hundred gallons of water in which are dissolved ten pounds of salt. A salt water solution containing one pound of salt per gallon begins to enter the tank at a rate of five gallons per minute. The well-mixed fluid leaves the tank through a pipe at a rate of five gallons per minute. Determine the amount of salt in the tank at time t minutes after the start of the process. Determine the ultimate amount of salt in the tank and the time at which the amount reaches ninety-five percent of the ultimate amount. Question 2 Solve the following differential equation and discuss the behavior of the solution: (1 + t 2 ) dy dt + 4ty = (1 + t2 ) 1, y(0) = 3. Question 3 Solve the following differential equation and discuss the behavior of the solution: dy + cos(t)y = 4 sin(t) cos(t), y(0) = 0. dt 6

Question 4 A bacterial population is governed by the logistic equation. Initially there are 1000 bacteria, increasing at a rate of 500 bacteria per hour. The limiting population is 20000 bacteria. Find the time for the bacterial population to reach 10000 bacteria. Question 5 Tank A initially contains one hundred gallons of water in which are dissolved 25 pounds of salt. A salt water solution containing 0.5 pounds of salt per gallon begins to enter the tank at a rate of six gallons per minute. The well-mixed fluid leaves the tank through a pipe at a rate of six gallons per minute and goes into tank B, which initially contains 50 gallons of pure water. The well-mixed fluid in tank B is drained through another pipe also at a rate of six gallons per minute. Determine the amount of salt in the tanks at time t minutes after the start of the process. Also determine the ultimate amount of salt in each tank. 7

Homework 4, due Friday 2nd February 2007 Read the first four chapters of the book up to the end of chapter four. Do the assigned book exercises to the end of section 4.5. Do the following problems to be handed in for grading. Question 1 Tank A, which has a capacity of two hundred gallons initially contains one hundred gallons of water in which are dissolved fifty pounds of salt. Tank B initially has five pounds of salt dissolved in forty gallons of fluid. Water containing 0.5 pounds of salt per gallon flows into tank A at a rate of eight gallons per minute, is well-mixed and exits the tank through a pipe to tank B at a rate of four gallons per minute. Finally the well-mixed fluid in tank B is drawn off at a rate of two gallons per minute. The process continues until tank A is full. How much salt is in each tank at that time? Question 2 A circuit contains an inductance of 10 henrys and a resistance of 30 ohms. The circuit has a driving electromagnetic force of 60 cos(2t) volts. Given that the initial current is zero discuss the subsequent behavior of the current in the circuit. Question 3 A particle of mass two kilos falls from rest under gravity, in a medium where the frictional retarding force (measured in Newtons) is four times the velocity (where the velocity is measured in meters per second). How long does it take for the particle to fall a distance of 100 meters? Question 4 A circuit contains an inductance of 30 henrys, a capacitance of 0.05 farads and a resistance of 50 ohms. There is no driving electromagnetic potential. Given that the initial charge in the system is 100 Coulombs and the initial current is zero amperes discuss the subsequent behavior of the current in the circuit. 8

Question 5 Solve the following differential equation and discuss the behavior of the solution: y + 8y + 16y = 72e 2t, y (0) = 0, y(0) = 12. 9

Homework 5, due Wednesday 7th February 2007 Read the first four chapters of the book up to the end of chapter four. Also read Chapter 6, section one. Do the assigned book exercises to the end of section 6.1 (not chapter 5, yet). Prepare for the exam 2/9/7: topics chapters 1 through 4. Do the following problems to be handed in for grading. Question 1 A circuit contains an inductance of 20 henrys, a capacitance of 0.01 farads and a resistance of 120 ohms and is driven by a constant voltage of 120 volts. Given that the initial charge in the system is 500 Coulombs and the initial current is zero amperes, discuss the subsequent behavior of the current in the circuit. Question 2 Solve the following differential equation and discuss the behavior of the solution as a function of time: Question 3 y + 6y + 25y = 160e 3t, y(0) = y (0) = 0. Solve the following differential equation and discuss the behavior of the solution as a function of time: Question 4 y 3y + 2y = 30 sin(3t) 410 cos(3t), y(0) = y (0) = 0. Find the solution of the following differential equation and discuss the behavior of the solution as a function of t: y + 9y = 15 sin(2t), y(0) = 2, y (0) = 6. Also describe a physical model that could lead to this differential equation. 10

Question 5 Use Euler s method with step size h = 0.1 to compute the first five iterations for the system: x = x y, y = y 3x, x(0) = 1, y(0) = 5. Plot your results. Also plot the exact solution [x, y] = 2e 2t [1, 1] e 2t [1, 3] and compare your results with the exact solution. 11

Homework 6, due Friday 23rd February 2007 Analyze your exam, compare with the solutions and make sure you can do all the problems. Read your textbook up to the middle of chapter eight. Do the assigned book exercises to the end of section 8.3 (not chapter 5, yet). Do the following problems to be handed in for grading. Question 1 Set up Euler s method for the system x = y, y = x, with initial condition (x, y)(0) = (1, 0) and arbitrary step size h as a matrix recursion. Compute the time 1 flow with h = 0.2, h = 0.1 and h = 0.05 and plot your results. Also compare with the exact solution. Discuss your results. Question 2 Solve the following system: d dt x v = 0 1 2 2 x v. (Hint: write out the system, show that v can be replaced in it by x and therefore v by x and then solve the resulting second order equation for x). The initial conditions are x(0) = 1, v(0) = 0. Plot the solution and describe the plot. Question 3 Find the general solution of the following system and discuss the resulting trajectories: u = v(u 2 + v 2 ), v = u(u 2 + v 2 ). (Hint: first compute the derivative of u 2 + v 2 ). 12

Question 4 Find the general solution of the following system and discuss the resulting trajectories: x = x 2 y, y = xy 2. (Hint: first compute the derivative of xy). Question 5 Consider the following predator-prey system: Find the equilibrium solutions. 10x = 2x xy, 10y = 3y + xy. Plot the phase plane, with several solutions superimposed. Discuss the qualitative behavior of the solution with the initial condition: (x(0), y(0)) = (3, 2). Find and plot the level surfaces of the solution: hint dy dx = y gives a x separable differential equation, which can be solved. In particular plot the level surface passing through the point (3, 2). Discuss the stability of the equilibrium solutions. 13

Homework 7, due Friday 2nd March 2007 Read your textbook up to the middle of chapter nine. Do the assigned book exercises to the end of section 9.3 (not chapter 5, yet). Do the following problems to be handed in for grading. Question 1 Consider the system: x = x(4 2x y), y = y(4 x 2y). Sketch the x and y null-clines and for each null-cline sketch the direction field of the system at various points of the null-clines. Identify the equilibria of the system Discuss and sketch the evolution of the system, for various initial conditions. Question 2 Solve the following matrix system, with the given initial condition and discuss the behavior of the solution: Question 3 X = AX, X = x y, A = 1 2 3 4, X(0) = 3 1. Solve the following matrix system, with the given initial condition and discuss the behavior of the solution: X = BX, X = x y, B = 2 1 1 2, X(0) = 1 1. 14

Question 4 Solve the following matrix system, with the given initial condition and discuss the behavior of the solution: X = CX, X = x y, Question 5 C = 4 1 4 0, X(0) = 4 4. Tank A contains one hundred gallons of water in which one hundred pounds of salt is dissolved. Tank B contains two hundred gallons of water in which fifty pounds of salt is dissolved. Seventy gallons per hour of well-mixed fluid flows out from tank A along a pipe into tank B. The well-mixed fluid in tank B flows along a second pipe back into tank A at a rate of twenty gallons per hour. The well-mixed fluid is also pumped out of tank B into a reservoir at a rate of ninety gallons per hour. Finally fifty gallons per hour of pure water is pumped into tank A and forty gallons per hour of pure water is pumped into tank B. Determine the amount of salt in each tank as a function of time. 15

Homework 8, due Friday 16th March 2007 Read your textbook up to the middle of chapter ten. Do the assigned book exercises to the end of section 10.3. Do the following problems to be handed in for grading. Question 1 Two nodes A and B are connected by a resistance R ohms, a capacitance C farads and an inductance L henrys in parallel. Denote by V volts the voltage drop across the capacitor and by I amperes the current through the inductance. Show that V and I obey the differential equations: V = V RC I C, I = V L. Consider the case that R = 1 ohms, C = 1 farad and L = 1 henry, with V 2 2 initially 10 volts and I initially zero. Solve the system and plot the voltage and current as a function of time. Question 2 Tanks A and B each contain 360 liters of brine. Pure water enters tank A from a tap at a rate of five liters per minute. Nine liters per minute of well-mixed fluid flows out from tank A along a pipe into tank B. The well-mixed fluid in tank B flows along a second pipe back into tank A at a rate of four liters per minute. The well-mixed fluid is also pumped out of tank B into a reservoir at a rate five liters per minute. Initially tank A contains sixty kilograms of salt, whereas tank B contains only pure water. Determine the amount of salt in each tank as a function of time and discuss your results. If the reservoir is initially empty, also determine the amount of salt in the reservoir as a function of time and discuss your results. 16

Question 3 Classify the phase portraits (sink, source, spiral, node, saddle, etc.) for the system X = AX, for the following cases: A = 2 1 2 4 A = 4 2 10 4 A = 4 2 2 4 Question 4 Consider the system: x = x(4 2x y), y = y(4 x 2y). Identify the equilibria of the system Find and solve the linearization of this system around each equilibrium point and discuss your results. In particular classify the nature of each equilibrium point. Question 5 Consider the system: x = x + y, y = y(1 x 2 ). Identify the equilibria of the system Find and solve the linearization of this system around each equilibrium point and discuss your results. In particular classify the nature of each equilibrium point. 17

Homework 9, due Wednesday 28th March 2007 Read your textbook up to the end of chapter ten. Do the assigned book exercises to the end of chapter ten. Do the following problems to be handed in for grading. Question 1 Classify the nature of the equilibria for the differential equation of a pendulum: x = sin(x) (begin by writing this a first order system). Question 2 Classify the nature of the equilibria for the following system: x = 4x(3 x) 3xy, y = y(1 y) xy. Also explain why a solution that starts in the first quadrant must remain there for all time. Question 3 Let f(t) = sin(3t). Determine the Laplace transforms of the following functions: f(t) f(2t) f (t) e t f(t) t 2 f(t). Question 4 3s + 10 Let a function f(t) have Laplace transform the function g(s) = s 2 + 6s + 8. Find the function f and determine the Laplace transform of f (t). 18

Question 5 By using Laplace transforms, solve the differential equation y 5y = e 3t, with the initial condition y(0) = 5. 19

Homework 10, due Friday 13th April 2007 Question 1 Using Laplace transforms, solve the following differential equation and discuss the behavior of the solution, with a graph; in particular discuss the sense in which the boundary conditions are obeyed by your solution: Question 2 y + 5y + 6y = 5 sin(t) + 15 cos(t), y(0) = 1, y (0) = 1. Let a function f(t) with domain the reals be given as follows: For t < 0 and for t > 4, f(t) = 0. For 0 t 2, f(t) = 3t 3. For 2 t 4, f(t) = 9 3t. Sketch the graph of f. Write a formula for f(t) using Heaviside functions and find its Laplace transform. If a function F (t) is periodic on the real line, with period 4, and F (t) = f(t) for 0 t 4, sketch the graph of F (t) and determine its Laplace transform. Question 3 Using Laplace transforms, solve the differential equation y + 4y = δ(t), with initial conditions y(0) = 2, y (0) = 4 and discuss the behavior of the solution with a graph. Question 4 Find the Fourier series for the function F (t) of question two above. Also, for 2 t 2, plot the first three partial sums of the Fourier series and the function F (t) on the same graph and discuss your results. 20

Question 5 Find the complex Fourier series for the function e 2t on the interval [ π, π]. Also plot the graph of the function e 2t and the first three partial sums of the Fourier series on the interval [ π, π] and discuss your results. 21

Homework 11, due Friday 20th April 2007 Question 1 Using Laplace transforms, solve the following differential equation and discuss the behavior of the solution, with a graph; in particular discuss the sense in which the boundary conditions are obeyed by your solution: Question 2 y + 5y + 6y = 4δ(t), y(0) = 1, y (0) = 1. Show that any solution u(t, x) of the first order partial differential equation u t = cu x obeys the wave equation u tt = c 2 u xx. Also show that any solution of the first order partial differential equation u t = cu x obeys the wave equation u tt = c 2 u xx. Hence, or otherwise show that if f and g are twice continuously differentiable functions then u = f(x ct) + g(x + ct) obeys the wave equation u tt = c 2 u xx. Finally if the initial values u(0, x) = a(x) and u t (0, x) = b(x) are given, find a formula for the functions f and g in terms of a and b. Question 3 Solve the equation u t = 2u x by separation of variables. Using the principle of linear superposition, find the solution with u(0, x) = e x + 3e 2x + sin(x). Discuss its behavior as a function of time t and space x. Question 4 Solve the heat equation u t = u xx, with initial condition: u(0, x) = sin(x)+sin(3x) and with boundary conditions u(t, 0) = u(t, π) = 0, for all t and discuss its behavior. Question 5 Find the solution of the heat equation u t = 4u xx, for the initial condition u(0, x) = cos(x) cos(2x) + sin(4x), with boundary conditions u(t, 0) = u(t, 2π) = 0 and discuss its behavior. 22