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
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 #11 Linear Systems
Goal: Investigate the behavior of a linear system of equations near its equilibrium point. Characterize the behavior in terms of the nature of the eigenvalues. Required tools: Matlab routine pplane , fplot; solutions of linear homog
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 #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
LAB #8 Numerical Methods
Goal: The purpose of this lab is to explain how computers numerically approximate solutions to dierential equations. Required tools: Matlab routine deld ; numerical routines eul, rk2, rk4; m-les. Discussion In this lab you will ap
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
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 #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
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 #3 The Existence and Uniqueness Theorems
Goal: Determine under what circumstances solution curves for a dierential equation x = f (t, x) can cross; examine when can a solution not exist and when there are multiple solutions; use Existence and Uniquene
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
Introduction to deld , pplane and fplot
It might be reasonable to expect that if two solutions of a dierential equation which start close together, then they should stay relatively close together. We can investigate the behavior of solutions using the rou
Lab 12 Expectations
Submit Plots: 1a, 1b, 2a, 2b, 3b, 3c, 4a, 5d
1.
a. Use pplane to plot several orbits for (*). What kind of solution does the
origin seem to be?
b. Plot these solutions on the square |x,y|<0.1
c. Let A be the 2x2 matrix for (*). Find th
Lab 11 Linear Systems
Expectations
1.
2.
3.
4.
5.
6.
7.
8.
Find the equilibrium points for (*).
Show that the eigenvalues are as shown in the lab.
Using fplot, plot y=x/4 for |x|<4 and |y|<4. Label the T and D axes.
a) Indicate on the plot the points for
Lab #10: SIR Model of a Disease
Expectations
1.
2.
3.
4.
5.
a) Use pplane to plot the system (*) for 0<t<4. Describe what happens
to S and I as t -> 4.
b) Approximate the maximum number of infected people in the first 4
months. Estimate when this occurs.
Lab Expectations Lab 9: A Predator-Prey Problem
1. Find the equilibrium points for the system.
2. Plot in pplane with correct scale (scale w/ origin at center and other equilibrium at
center of first quadrant)
What is the significance for the populations
Lab 8 expectations:
(1): Give the slope of the third line segment, and SHOW YOUR WORK!
(2): (Graph needed) Compute the slopes of the first two Euler lines, and
give the answers. Confirm that the slope is in fact the average of the two
Euler slopes. Then f
Lab 07 Expectations
Submit Plots: 1, 3, 4, 6
1. - Pplane plot of v vs. y, and pplane plot of y vs. t.
- Estimate the amplitude and period from the graph.
- Would you consider this a slow or fast oscillation?
2. - Find the general solution.
- Find the part
Lab 06 Expectations
3 plots required: best estimate for y1, best estimate for y2, pplane plot from #4.
1.
2.
3.
4.
Determine which of the equation forms a better model and why.
For the better equation, determine the best approximate values of the paramete
LAB #5 NOTES
PROBLEM 1
Type in the MATLAB code as shown. Print off the graph. To highlight each datapoint, you could
use the command: plot(T,A,
-*);
PROBLEM 2
Solve the separable ODE (
dP
= r ) using P(0)=379 and P(1) = 423. To predict the population in
d
Lab 05 Expectations
Submit Plots: 5
t=23 corresponds to the year 2020.
1. Plot actual data, this will be printed in number 5.
2. Find unique solution to (*) using P(0)=379 and P(1)=423. SHOW WORK. Predict population in
2020.
3. Find unique solution to (*)
Lab 4 Notes
Last Updated: 1:45 PM 2/5/2008
In Maple 10, make sure you use the command line text instead of the math type text by clicking on the
following"text" and "[>" buttons.
Lab 04 Expectations
Submit Plots: 4, 5
1. Solve each initial value problem.
2. a. Seed = 10. Use dsolve in Maple to find f and g.
b. Substitute the equations into (*) to make sure these functions satify the differential equation.
c. Substitute h(x) into t
Lab 03 Expectations
Submit Plots: 1, 2, 4 (2 graphs)
1. Dfield7 plot of your chosen differential equation with 4-6 solutions and explanation about the
theorems.
2. a) Dfield7 plot of the given differential equation with 4-6 solutions. What kind of curves
Lab 02 Notes
Last Updated: 4:57 PM 8/25/2008
3: To convert from the units of v (dy/ds) to mi/sec (dx/dt):
dx/dt = 4000*dy/dt = 4000*dy/ds*ds/dt = 4000*v*ds/dt = 4000*sqrt(0.0061/4000)*v
Lab 02 Expectations
Submit plots: pplane 3, 4, 5a, 5b, 6; fplot 6.
Units in this lab is important. Make sure you express your answers in the correct units.
1. Nothing is required. Note: this will work if you substitute t for y in dfield.
2. Make the graph
Lab 01 Notes
The central concept in this lab is newton's law of cooling. Check out the wikipedia article.
Also this solved problem that is similar to this lab, or this.
An applet.
An experiment you can do at home.
Lab 01 Expectations
Make sure all plots are labeled well, and show the value that you were asked to find to the correct number of
decimals.
Submit plots: 2, 3, 5, 8
1. Show that k = .39 using the two equations that are given.
2. Approximate the number of