MA 266 FINAL EXAM INSTRUCTIONS May 2, 2005 NAME INSTRUCTOR
1. You must use a #2 pencil on the marksense sheet (answer sheet). 2. If the cover of your question booklet is GREEN, write 01 in the TEST/QUIZ NUMBER boxes and blacken in the appropriate sp
Computer Project 1. Nonlinear Springs
Goal: Investigate the behavior of nonlinear springs. Tools needed: ode45, plot Description: For certain (nonlinear) spring-mass systems, the spring force is not given by Hooke's Law but instead satisfies Fspring = ku
Computer Project 3. Predator-Prey Equations
Goal: Investigate the qualitative behavior of a nonlinear system of differential equations. Tools needed: pplane8 Description: A farmer has ladybugs and aphids in her fields. The helpful ladybugs (predator) eat
Supplementary Problems
A. For what value(s) of A, if any, will y = Ate2t be a solution of the dierential equation
2y + 4y = 3e2t ? For what value(s) of B , if any, will y = Be2t be a solution?
B. Using the substitution u(x) = y + x, solve the dierential e
2.2 Separable Equations. Homogeneous
equations
Evgenia Malinnikova, Purdue University
January 19, 2016
Evgenia Malinnikova, Purdue University
MA26600, Lecture 4
Separable equations
The standard form of the general first order equation is
y 0 = f (x, y )
M
2.4
17)
Y goes to 3 if y0 > 0; y=0 if y0=0;y goes to infinity if y0 < 0
22)a)
As we can see this differential equation clearly has the solution of y=1-t, but only when t is greater than
or equal to 2, as shown in the directly field. However, y=-t^2/4 is a
Project 2
1)
function xp=F(t,x)
%xp=zeros(2,1); % since output must be a column vector
xp(1)=x(2);
xp(2)= -(4 * x(2) - (5 * x(1) + cos(omega* t);
xp=xp';
end
[t,x]=ode45('F',[0,20],[0,1])
plot(t,x(:,1)
*The variable omega in the function xp is the one tha
2.5
4) dfield of y prime = e^y-1
Graph shows instability at the critical point y=0
Euler Method
Supplementary Problems
G) b) euler method on matlab to estimate y prime = -2y + e^-t y(0)=1 by 0.05
Y(1) = e^-1 = 0.367879
Bounds: [0.317879 ,0.417879]
N=3
> [
TEST 2, MATH 262
Kapitanov
SPRING 2013
NAME ( cfw_brag
SECTION(Cii-c1e one): 041(11230 MWF);05 1(12130 MWF)
I pledge I have neither received nor given unauthorized help on this exam.
Pledge:
Instructions: 10 Problems total, 10 pts each. All involve multip
Handwritten HW #19 21
3.7
9) plot u v t
28) Plot u v t and u prime v t on same axes then plot u prime v u parametrically
The direction of motion of the phase plot is clockwise.
Same Plot
Parametric
3.8
7) plot the graph of the solution
3.1
19) plot the solution from 0 to 2 and find the minimum
Minimum y value: 1 @ t = 0.693 = ln(2)
3.3
23) when t>0 find the first time at which u(t) = 10 absolute value
t = 10.76 approximately
2.3 Modeling with first order equations
Evgenia Malinnikova, Purdue University
January 26, 2016
Evgenia Malinnikova, Purdue University
MA26600, Lecture 5
Gravitation force: Example 0
A ball of mass m = 0.1 kg is dropper from a tower of 100 m high.
The ini
2.6 Exact equations/2.7 Eulers method
Evgenia Malinnikova, Purdue University
February 4, 2016
Evgenia Malinnikova, Purdue University
MA26600, Lecture 8
Exact equations
Suppose that for a first order differential equation
M(x, y ) + N(x, y )y 0 = 0
one can
2.5 Autonomous equations
Evgenia Malinnikova, Purdue University
February 2, 2016
Evgenia Malinnikova, Purdue University
MA26600, Lecture 7
Autonomous equation
A first order differential equation of the form
y
= f (y )
t
is called autonomous (the right-han
2.4 Linear and nonlinear first order equations
Evgenia Malinnikova, Purdue University
January 28, 2016
Evgenia Malinnikova, Purdue University
MA26600, Lecture 6
Bernoulli equation
The following class of nonlinear equations can be solved by a
substitution:
Classification of Differential equations
Evgenia Malinnikova, Purdue University
January 14, 2016
Evgenia Malinnikova, Purdue University
MA26600, Lecture 2
Reminder
Last time we learned
I
direction fields
I
integral curves
I
general solutions
I
initial val
2.8 The existence and uniqueness theorem/
Review ch1-2
Evgenia Malinnikova, Purdue University
February 9, 2016
Evgenia Malinnikova, Purdue University
MA26600, Lecture 9
Nonlinear equations: existence and uniqueness
Theorem
f
If the functions f (t, y ) and
Charya Ratwatte MA
266 Project 2
1) Use ode45 (and plot routines) to plot the solution of () with Q(0) = 0 and Q0 (0)
= 0 over the interval 0 t 80 for = 0, 0.5, 1, 2, 4, 8, 16.
Graphs are in respective order of omega-listed values.
Charya Ratwatte MA
266
3.1 Second order linear homogeneous equations
with constant coefficients
Evgenia Malinnikova, Purdue University
February 16, 2016
Evgenia Malinnikova, Purdue University
MA26600, Lecture 10
First and second order linear homogeneous equations
Remind that a
6.5
14)a)plot the solution to the initial value problem
b) find the time t1 where I reaches its maximum value and the y1 of it
Maxmimum Point marked by the blue dot above
T = 2.361
Y = 0.712
Marco Manoppo
Q1.
=0
= 0.5
MA 266 Project 2
=1
=2
=4
=8
= 16
Q2.
, A() = 0
0, A() = 2
Thus, as time increases to infinity the displacement will approach zero. As for time, when it goes
to zero the displacement will be 2.