WORKSHEET #4: MATH 317
ORDINARY DIFFERENTIAL EQUATIONS COMPUTER LAB
Two computer labs ago you wrote computer code that implemented Eulers Method
yi+1 = yi + hf (xi , yi ),
i = 0, 1, 2, . . .
for approximately solving the IVP
dy
= x + y, y(0) = 1,
(1)
dx
o
WORKSHEET #12: MATH 317
ORDINARY DIFFERENTIAL EQUATIONS COMPUTER LAB
Solving Linear Systems Using MATLAB
During the past several labs, we have studied the predator-prey model for the interaction of
two species x and y given by
dx
= ax Ay,
dt
dy
= Bx by.
d
WORKSHEET #13: MATH 317
ORDINARY DIFFERENTIAL EQUATIONS COMPUTER LAB
Using MATLAB to solve various ODEs.
In this our nal lab, we are going to use, exclusively, MATLABs built in ODE solvers to
solve a variety of ODEs.
Problem 1: First, we consider the logi
WORKSHEET #10: MATH 317
ORDINARY DIFFERENTIAL EQUATIONS COMPUTER LAB
Solving Linear Systems Using MATLAB
Last time we computed the eigenvalues and eigenvectors of the 2 2 matrices below using
MATLAB.
1. A =
12 7
7 2
2. A =
2 1
0 3
3. A =
3 2
5 1
.
.
.
Pro
WORKSHEET #10: MATH 317
ORDINARY DIFFERENTIAL EQUATIONS COMPUTER LAB
Eigenvalues and Eigenvectors of a Matrix Using MATLAB
Computing eigenvalues and eigenvectors of a matrix using MATLAB is quite straightforward. From yesterdays class we had several 2 2 m
Math 311-QUIZ 7
NAME:
Show all work to receive partial credit!
1. Find the solution of the initial value problem
y y y +y =0
y(0) = 0, y (0) = 1, y (0) = 0,
given that the characteristic equation ( 1) as a factor. Note that youll need to use
long division
Math 311-QUIZ 10
NAME:
Show all work to receive partial credit!
1. Find the eigenvalues and eigenvectors of the matrix
A=
4 13
2 6
(1)
2. Verify that
(t) =
e4t
1 4t
e
2
et
1 t
e
3
is a fundamental matrix solution of
x =
1 6
1 2
x
by showing that the colum
WORKSHEET #1: MATH 317
ORDINARY DIFFERENTIAL EQUATIONS COMPUTER LAB
Course Goal: To be introduced to and to gain a basic understanding of numerical ODEs.
Worksheet Goal: Today we will spend some time introducing ourselves to the main software package
that
Math 311-QUIZ 11
NAME:
Choose one of the two problems below as your quiz problem. The other
problem will then be viewed as extra credit.
1. Use the method of determinants to nd the solution of the linear system
x
y
= 4x 3y,
= 5x 4y,
x(0) = 1,
y(0) = 1.
2.
Math 311-QUIZ 9
NAME:
Show all work to receive partial credit!
1. Consider the matrix
1 1 1
A = 1 2 2
1 4 4
(a) Show that A is invertible by showing that det(A) = 0
(b) Use Gaussian elimination to nd A1 .
(1)
2. On the last quiz, you were asked to expres
Math 311-QUIZ 8
NAME:
Show all work to receive partial credit!
1. Write y y 4y + 4y = 0 as a system of rst order equations.
2. Suppose that the current I of an RLC circuit is governed by the ODE
I + 4I + 4I = 50 sin 5t.
Find the general solution of this e
Math 311-QUIZ 6
NAME:
Show all work to receive partial credit!
1. If y1 is a solution of
y + a(x)y + b(x)y = 0,
a second solution is given by
y2 (x) = y1 (x)
e a(x)dx
dx.
y1 (x)2
(1)
Given that y1 (x) = xe3x is a solution of
y 6y + 9y = 0,
use (1) to nd a
Math 311-QUIZ 3
NAME:
Show all work to receive partial credit!
1. Use separation of variables to nd the solution of the IVP
dy
= y(1 cos x),
dx
y(0) = 1.
2. Use the method of integration factors to solve the rst-order, linear IVP
dy
+ 5y = x,
dx
y(0) = 1.
Math 311-QUIZ 1
NAME:
Show all work to receive partial credit!
1. A ball is thrown upward with an initial velocity v0 = 7.7 meters per second from the
top of a building h0 = 56 meters high. Assume that the height of the ball in meters at
time t is governe
Math 311-QUIZ 4
NAME:
Show all work to receive partial credit!
1. Consider the equation
y cos(xy)dx + x cos(xy)dy = 0
(a) Verify that (1) is an exact dierential equation.
(b) Solve (1).
(1)
2. Consider a tank holding 50 gallons of water in which are disso
Math 311-QUIZ 2
NAME:
Show all work to receive partial credit!
1. Consider the dierential equation
x2 y 5xy + 9y = 0.
(a) What is the order of (1)?
(b) Show that y(x) = 2x3 ln x is a solution of (1) for x > 0. Work carefully!
(1)
2. Determine which of the
WORKSHEET #6: MATH 317
ORDINARY DIFFERENTIAL EQUATIONS COMPUTER LAB
PROBLEM 1: Use the taylor2.m from the web-page to solve the IVP
y = cos x sin y + x2 ,
y(1) = 3.
(1)
Use h = 0.1 and nd an approximate solution on [1, 1]. Plot your solution together with
WORKSHEET #5: MATH 317
ORDINARY DIFFERENTIAL EQUATIONS COMPUTER LAB
PROBLEM 1: Use the Eulers method code from the web-page to solve the IVP
y = cos x sin y + x2 ,
y(1) = 3.
(1)
Use h = 0.1 and nd an approximate solution on [1, 1]. Then plot it. This will
Math 311 Fall 2012
Test 1 (100 points)
problem
1
2
3
4
5
total
points
30
15
15
20
20
100
Name:
score
1. (30 pts) Solution Techniques
a) One of the following is an exact ODE, the other is not. Determine this and nd the general
solution (in implicit form) t
Math 311-EXAM 2 REVIEW
Place: The usual.
Time and Date: 9:10-10am, 11-09-05.
Sections That Will Be Covered:
1. The sections that we covered in Chapter 3 were 3.2-3.8. The main topics are reduction
of order (3.2); linear, constant coecient, homogeneous ODE
Math 311-Final Exam
Place: The usual; Time and Date: 8:-10am, Thursday December 15, 2005.
Cheat Sheet: You can have one side of an 8 1 11 sheet of paper lled with formulas
2
for part I. If you are also taking part II, you can ll both sides with formulas.
Math 311-EXAM 1 REVIEW
Place: The usual.
Time and Date: 9:10-10am, 9-28-05.
Sections That Will Be Covered:
1. The sections that we covered in Chapter 1 were 1.1, 1.2, 1.3, and 1.4. Note that no
problem from 1.4 (Eulers Method) found its way onto a quiz, b
Math 311-EXAM 1, Fall 04
NAME:
Show all work to receive partial credit!
1. Consider the dierential equation
dy
= x y,
dx
(a) Find the function y that solves (1).
y(1) = 2.
(1)
(b) What is y(2) in decimal form?
(c) Use Eulers method with h = 0.25 to approx