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
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
MATH 266 - WEEK 2 PROBLEMS - MATH MODELS
Contents
1.
Mixing type : HW problem 1 4
2
1.1. cin (t) constant
2
1.1.1.
Same rate of liquid entering and draining
3
1.1.2.
Dierent rate of liquid entering and draining
3
1.2.
2.
With variable (t)
4
Gravitation ty
MATH 266 - WEEK 2 PROBLEMS - MATH MODELS
Contents
1.
Mixing type : HW problem 1 4
2
1.1. cin (t) constant
2
1.1.1.
Same rate of liquid entering and draining
3
1.1.2.
Dierent rate of liquid entering and draining
3
1.2.
2.
2.1.
With variable (t)
4
Gravitati
Math 266
Midterm #2 Part B
Your Name
Spring 2013
Your Signature
This exam is closed book.
The Laplace transform table is provided.
Calculators are not allowed.
In order to receive credit, you must show your work. Do not do computations in your head.
I
MATH 266 - WEEK 4
Contents
1.
Classication of Dierential Equations
1
2.
A statement on existence and uniqueness
2
2.1.
If remove continuity condition
2
2.2.
If we only have continuous but not C 1
2
3.
Dierence between linear vs nonlinear
3.1.
Interval of
Name:
MA 266, Fall 2009, Quiz 6
(1) (10 points) A certain vibrating system satises the equation
u + u + u = 0,
0.
Find the value of the damping coecient for which the quasi period of the damped
motion is 50% greater than the period of the corresponding u
function multigraf(arg)
% MULTIGRAF
%
%
%
%
%
%
%
enables putting up to six MATLAB figures into a single
figure. To use it, simply enter MULTIGRAF at the MATLAB
prompt. This will open a figure with six available slots.
Simply make a choice for each slot.
MATH 266 - WEEK 4
Problem 0.1 (Section 2.4 Problem 6).
(ln y )y + y = cot t
y (2) = 3
(1)
Find an interval in which the solution of the given initial value problem is certain
to exists
cot t
1
y=
ln t
ln y
For ln t is only dened on (0, ) with
y+
ln t = 0
MATH 266 - 2.6
Contents
1.
In the language of vector eld
1
1.1.
An example of a domain not simply connected
2
1.2.
Review
2
2.
Another way to nd the potential
3
2.1.
Denite integration
3
2.2.
Indenite integration
4
3.
In the language of dierential forms
6
Dierential Equations Practice: 2nd Order Linear: Application: Vibrations
Page 1
Questions
Example (3.7.1) Determine 0 , R, and so u = 3 cos 2t + 4 sin 2t = R cos(0 t ).
Example (3.7.6) A mass of 100g stretches a spring 5cm. If the mass is set in motion fr
MATH 266 - SAMPLE PROBLEM FROM SECTION 2.1
Contents
1.
Transition behavior near
2
2.
Transition behavior near 0
3
3.
Finding limit at
5
4.
Other problems
6
5.
Find tangency
7
1
1. Transition behavior near
Problem 1.1 (Problem 23).
3y 2y = et/2
y (0) =
MATH 266 - FIRST ORDER AUTONOMOUS EQUATION
Contents
1.
Equilibrium solutions
1
2.
General behavior of y = F (y )
2
3.
Autonomous equation of type y = y 2 + ay + b
3
3.1.
Type 1 : b
a2
>0
4
3
3.2.
Type 2 : b
a2
<0
4
3
3.2.1.
Preparation calculation
4
3.2
Name:
MA 266, Fall 2009, Quiz 1
(1) (10 points) Evaluate the following integral:
x2 ex dx.
Solution:
ex
x2 ex dx = x2
u
u
dv
ex 2x dx .
v
= x2 e x 2
v
du
x ex dx
u
= x2 ex 2xex + 2
dv
ex dx
= x2 ex 2xex + 2ex + C
(2) (10 points) Evaluate the following in
Name:
MA 266, Fall 2009, Quiz 3
(1) (10 points) Determine the solution of the following initial value problem. Determine the
interval in which the solution is dened.
y = (1 2x)/y,
y (1) = 2.
Solution: This equation is separable, since it can be written as
Name:
MA 266, Fall 2009, Quiz 2
(1) (8 points) Determine whether each of the following equations is linear or nonlinear. (+2
points for each correct answer, 2 points for each incorrect answer.)
Additionally, determine the order of each equations. (+2 poin
MATH 266: THIRD MIDTERM
(1) (10 points) Find the general solution of
y (4) 3y
+ 3y y = 0
In order to nd the general solution we have to nd the characteristic
equation rst r4 3r3 + 3r2 r = 0. Factoring we get r(r 1)3 = 0 and
we have the following roots: r
MA 26600
PRACTICE FINAL EXAM
1. The general solutions to ty y = t2 et is
A. y = et + c
B. y = tet
C. y = cet + t
D. y = tet + ct
E. y = tet + t
2. The general solution of
3y 2x
dy
=
is
dx
y
A. |y x| = c|x|(y 2x)2
B. |y x| = c(y 2x)2
C. (3y 2x)2 = c + |x|1
MATH 266: FIRST MIDTERM
(1) (8 points) Determine the order of each of the dierential equations; also
state whether the equation is linear or nonlinear.
(a) y
+ 2y + 3y + y = t2
(b) y + sin(t y ) = sin(t)
LINEAR - 3RD ORDER
NONLINEAR - 2ND ORDER
(c) y ty 2
MATH 266: SECOND MIDTERM
(1) (15 points) Given the initial value problem
y + 2y = 1 + t
with y (0) = 0, use Eulers method to approximate y (1) with steps of length
1/3.
y = 1 + t 2y
Then y (0) = 1 + 0 2(0) = 1, y (1/3) y (0) + 1/3y (0) = 0 + 1/3 = 1/3,
an
Section 3.8 and 3.9
MA266- 2009
Forced vibrations
Without damping: Consider mu + ku = F0 cos(t). The homogeneous solution uc is
c1 cos(0 t) + c2 sin(0 t)
k
2
where 0 = m .
Case 1: = 0 . In this case, the general solution u(t) is
F0
cos(t).
2)
u(t) = c1 c
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
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
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