x ' = t x3 4
-4 -2 0 2 4 t 6 8 10
x ' = 2 x/t 4
-4 -2 0 2 4 t 6 8 10
Parabola, they appear to cross at (0,0). b. tx' = 2x dx/x=2dt/t lnx = 2lnt+c e^(ln(t^2)+c) = x t^2*e^c = x x = ct^2 when t =0, x=0 . It's n
LAB #1 CSI - Time of Death
Goal: Approximate parameters in a dierential equation using calculus; approximate the solution of a dierential equation using direction elds; solve the dierential equation; compare approximate solutions to true solutions. Requir
LAB #2 Escape Velocity
Goal: Determine the initial velocity an object is shot upward from the surface of the earth so as to never return; illustrate scaling variables to simplify dierential equations. Required tools: deld and pplane ; second order dierent
MA 366 Spring 2011 Assignments
For Wednesday 1/12: Read 1.21.3. Do: p. 25: 7, 9, 16 p. 360, The answer to Exercise 8 is given in the back of the text. Substitute the given functions x1 and x2 into the the system to show that they do solve the system. Show
SAMPLE MIDTERM EXAM #2
This exam is to be done in 50 minutes in one continuous sitting. Collaboration is not allowed,
but you may use your own personal notes; the lecture notes posted on the course web site;
homework solutions posted on the
LAB #10 SIR Model of a Disease
Goal: Model a disease and investigate its spread under certain conditions. Use graphs generated by pplane (and its many options) to estimate various quantities. Required tools: Matlab routine pplane and its graphing options.
LAB #5 Population Models
Goal: Compare various population models for the population of New York over the last 200 years. Required tools: Matlab routines plot, norm, fplot; separable dierential equations. Discussion This lab compares three models of popula
MA 36600 FINAL EXAM SOLUTIONS
(a) Sketch the direction eld for the dierential equation y = 1+y 4 . Are there any equilibrium solutions?
(b) Sketch the direction eld for the dierential equation y = 1y 4 . Are there any equilibrium solut
LAB #12 Linearization
Goal: Investigate the local behavior of a nonlinear system of dierential equations near its equilibrium points by linearizing the system. Required tools: Matlab routine pplane ; eigenvalues and eigenvectors. Discussion In the last la
LAB #6 The Swaying Building
Goal: Determine a model of the swaying of a skyscraper; estimating parameters Required tools: Matlab routines pplane , ode45, plot; M-les; systems of dierential equations. Discussion Modern skyscrapers are built to be exible. I
LAB #4 First Order Linear Differential Equations
Goal: Introduction to symbolic routines in Maple to solve dierential equations; dierences in linear and nonlinear dierential equations; solutions to homogeneous equations; particular solutions Required tool
LAB #7 Resonance
Goal: Observe the phenomenon of resonance; nd numerical approximations of solutions to non-autonomous systems of dierential equations. Required tools: Matlab routines pplane , ode45 ; m-les; systems of dierential equations. Discussion Ass
MA 36600 MIDTERM #1 SOLUTIONS
Problem 1. For a xed real number r, solve the initial value problem
y + r y = r,
y (0) = 0.
Solution: First we nd the general solution to the dierential equation. Multiply both sides by the
integrating factor (t) = ert :
LAB #9 Predator-Prey Problems
Goal: Investigate the interaction of species via a particular predator-prey problem. Required tools: Matlab routines pplane , deld and fplot. Discussion You will examine a predator-prey problem that has historical roots as no
MA 36600 HOMEWORK ASSIGNMENT #6 SOLUTIONS
Problem 1. Sec. 3.2, pg. 156; prob. 31
In each of Problems 15 through 18 nd the Wronskian of two solutions of the given dierential equation
without solving the equation.
x2 y + x y + x2 2 y ,
MA 36600 HOMEWORK ASSIGNMENT #5 SOLUTIONS
Problem 1. Sec. 3.1, pg. 144; prob. 21
Solve the initial value problem y y 2 y = 0, y (0) = , y (0) = 2. Then nd so that the solution
approaches zero as t .
Solution: First we nd the general solution to the dieren
MA 36600 HOMEWORK ASSIGNMENT #4 SOLUTIONS
Problem 1. Sec. 2.7, pg. 110; prob. 8
In each of Problems 5 through 10 draw a direction eld for the given dierential equation and state
whether you think that the solutions are converging or diverging.
y = t y + 0
MA 36600 HOMEWORK ASSIGNMENT #3 SOLUTIONS
Problem 1. Sec. 2.5, pg. 88; prob. 3
Problems 1 through 6 involve equations of the form dy/dt = f (y ). In each problem sketch the graph of
f (y ) versus y , determine the critical (equilibrium) points, and classi
MA 36600 HOMEWORK ASSIGNMENT #9 SOLUTIONS
Problem 1. Sec. 7.1, pg. 359; prob. 7
Systems of rst order equations can sometimes be transformed into a single equation of higher order.
Consider the system
x1 = 2 x1 + x2 ,
x2 = x1 2 x2 .
(a) Solve the rst equat