Xiaoyu Feng
MA366 Lab1
Problem (1)
From the given equation and the differential equation(*), we get
T(0)= 76.4
T(1)=73.9
T(0)= -k(T(0)-70)=(T(1)-T(0)/(1-0)=-2.5
Then, -k(76.4-70)= -2.5
k = -2.5/6.4=0.391
Problem(2)
When the victim died, t approximately eq
Qianyu Deng
MA366 Lab#3
M 9:30-10:20
Problem(1)
According to the picture, no two solutions intersected so the solutions are unique in a rectangle.
Therefore, it holds the Existence and Uniqueness Theorem by definition.
Problem(2)
(a)
The curves seem to be
Qianyu Deng
MA366 Lab#7
M9:30-10:20
Problem(1)
y+0.25y=0
The period is approximately 16.
Problem(2)
Problem(3)
I think the oscillation would be faster since if the string becomes stiffer, k should be greater, and
the frequency would be greater, thus the p
Qianyu Deng
MA366 Lab#2
M 9:30-10:20
Problem(3)
When y=0, v=1, there exists maximum finite height x = 1 * 4000 = 4000
When y=0,v=0.5, there exists masimum finite height x = 0.142*4000=568
V1 = 1.46
V2 = 0.61
dx1/dt= 4000*sqrt(0.0061/4000)*1.46= 7.212
dx2/
Numerical Methods & .m Files
In order to use Matlab routines for the Euler, Improved Euler or Runge-Kutta Methods, you will need the files eul.m, rk2.m or rk4.m, respectively. These files are already present on all ITaP machines as standard software. (If
Student Name:
Answers
MA-366, Spring 2004, Final Exam
Note: You may use scratch paper. All solutions and answers must be on those sheets in the space provided. If the space provided is not enough, continue on the back or attach additional sheets (well org
Qianyu Deng
MA366 Lab#6
M 9:30-10:20
This solution of A best fits y1 where k=0, q=0.285, v0=0.175, p=0.008.
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
0
5
10
15
Compare solution of A to the y2 data.We have the picture below.
20
25
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
Qianyu Deng
MA366 Lab#12
M9:30-10:20
Problem(1)
(a)
x ' = - 3 x + sqrt(2) y
y ' = sqrt(2) x - 2 y
10
8
6
4
y
2
0
-2
-4
-6
-8
-10
-10
-8
-6
-4
The origin seems to be a sink point.
(b)
-2
0
x
2
4
6
8
10
x ' = - 3 x + sqrt(2) y
y ' = sqrt(2) x - 2 y
0.1
0.08
MA 36600 LECTURE NOTES: FRIDAY, APRIL 10
Systems of Linear Equations
Systems of Linear Dierential Equations. We return to the linear system
x
1
x
2
x
m
=
=
.
.
.
p11 (t) x1
p21 (t) x1
+
+
= pm1 (t) x1
p12 (t) x2
p22 (t) x2
+
+
.
.
.
+ pm2 (t) x2
+
Usin
MA366 Final
Last Name:
First Name:
Show all work. A correct answer without supporting work is worth NO credit! (Some calculators can solve differential equations.) There should be no "hard" integrals, unless you mess up somewhere. If this happens, just le
O with LinearAlgebra : Problem 3 4, 1, 0 , K 1, 1 , 1, 1, 2 2, O A d Matrix A :=
; 4 1 0 (1) 1 1 2
K 1 1 2
Im Maple I denotes the imaginary number i. Hence we denote the identity matrix by II. O II d DiagonalMatrix 1, 1, 1 1 0 0 II := O B d A K3$II 1 B :=
MA366 Sathaye
Final Thoughts
First, some general advice. Review notes (class and on line), quizzes and old exams. You will be allowed three sheets of notes to be used during the exam. Be sure to have these hand written or typed, but not Xerox copies of pr
Student Name:
Answers
MA-366 Spring 2003, Final Exam
Instructor: P. Stefanov May 6, 2003
Note: You may use scratch paper. All solutions and answers must be on those sheets in the space provided. If the space provided is not enough, continue on the back or
Numerical Methods & .m Files
In order to use Matlab routines for the Euler, Improved Euler or Runge-Kutta Methods, you will need the files eul.m, rk2.m or rk4.m, respectively. These files are already present on all ITaP machines as standard software. (If
4
MA 36600 LECTURE NOTES: MONDAY, APRIL 13
When yk = xk (k) we use the fact that x =
ip
n
dW
d
=
( )
xk (k)
dt
dt
j
pij (t) xjp to nd
k=1
=
n
i=1
=
n
n
i=1 j =1
n
n
dxi (i)
( )
xk (k) =
pij (t)
( ) xj (i)
xk (k)
dt
i=1 j =1
k=i
pij (t) W x(1) , . . .
MA 36600 LECTURE NOTES: MONDAY, APRIL 13
3
It suces then to show that x(c) = c for some constant c. Consider an initial value problem in the form
d (c)
x = P(t) x(c) ,
x(c) (t0 ) = x0 .
dt
According to the Existence and Uniqueness Theorem, we know that a
MA 36600 LECTURE NOTES: MONDAY, JANUARY 12
Course Information
Instructor: Edray Goins. Oce: MATH 612. Extension: 4-1936. E-Mail: [email protected]
Oce hours will be on Thursdays from 10:00 AM through 12:00 PM.
Meeting Times: The class will meet Mo
2
MA 36600 LECTURE NOTES: MONDAY, APRIL 13
Then the general solution to the nonhomogeneous equation is in the form
x(t) = c1 x1 (t) + c2 x2 (t) + X (t)
where c1 and c2 are constants. In terms of matrices, we see that the general solution is in the form
x(
2
MA 36600 LECTURE NOTES: MONDAY, JANUARY 12
Say that we have an object of mass m, perhaps measured in kilograms. Also, say that x = x(t) is the
position of an object, perhaps measured in meters, at time t, perhaps measured in seconds. Then
dx
v=
=x
dt
is
Hillary and Barack ECE302Fall2008sanghavi - Rhea
https:/projectrhea.org/rhea/index.php/Hillary_and_Barack_EC.
Hillary and Barack ECE302Fall2008sanghavi - Rhea
The Problem: Hillary and Barack have a date, but both are late; each arrives upto 1 hour late. I
Math 366: Elementary Differential Equations
Text: Instructor: Office: E-mail: Elementary Differential Equations, ninth edition by Boyce and DiPrima Richard Penney 822 Mathematics Building Phone: 49-41968 [email protected]
Procedures Homework will be due
MA366 Sathaye
Solving for yp
1. Practice for New Formulas Here f (D) is any polynomial in D with constant coefficients. Exponential Slide. f (D)eat v = eat f (D + a)v. General Product Rule of Leibnitz Dn (vw) = vDn (w) + n D(v)Dn-1 (w) + n D2 (v)Dn-2 (w)
Phase Portraits - pplane7
The routine pplane7 is already loaded on all ITaP machines as standard software. (If you are using your own copy of Matlab you may need to download pplane7 from http:/math.rice.edu/dfield) You may also access Matlab (Matlab 7) t
ode45 - Differential Equation Solver
This routine uses a variable step Runge-Kutta Method to solve differential equations numerically. The syntax for ode45 for first order differential equations and that for second order differential equations are basical
MA366 Sathaye Brief Notes on 4.1, 4.2
Notes on Chapter 4,7
1. We have already studied second order linear equations in great detail. Fortunately, our methods naturally extend to higher order equations without much modification. We give a brief summary of
MA366 Sathaye
Fast methods for some chapter 7 problems.
We show some faster methods to quickly get a fundamental matrix for linear systems of differential equations. Two variable systems. 1. Basic formulas. We consider a system X = AX where A is a 2 2 mat
MA366 Sathaye
Notes on Chapter 7 continued Continued Notes on early sections of Chapter 7.
1. Complex Eigenvalues(7.6). We now analyze a system DX = AX where A is an n n matrix with constant (real) coefficients where not all eigenvalues are real. It is us
MA366 Sathaye
Notes on Chapter 3 Sec. 4-8
1. Reduction of Order. 3.4. Note that the topic of repeated roots of the characteristic equation was already discussed earlier. Here we consider a linear order 2 equation f (D)y = 0 and assume that we already know
MA366 Sathaye
Notes on Chapter 3 Sec. 1-3
1. Notation. We shall use the following notation which is more convenient than those in the d book. For the derivative dt (y) = y we prefer the simpler notation Dt (y) or simply D(y) if the independent variable is
MA366 Sathaye
Notes on Chapter 3 Sec. 1-3
1. Notation. We shall use the following notation which is more convenient than those in the d book. For the derivative dt (y) = y we prefer the simpler notation Dt (y) or simply D(y) if the independent variable is