1.
x ' = t x3 4
3
2
1
x
0
-1
-2
-3
-4 -2 0 2 4 t 6 8 10
2. a.
x ' = 2 x/t 4
3
2
1
x
0
-1
-2
-3
-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
MA 36600
GOINS
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
Problem 1.
(a) Sketch the direction eld for the dierential equation y = 1+y 4 . Are there any equilibrium solutions?
Explain.
(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 :
y +r
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 ,
Bessels equation
Solu
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