Midterm 1 Solutions
Math 528
October 19, 2012
Computers are not allowed for this test. Show enough work that I can follow
your logic in case you make a mistake and deserve partial credit.
1a. (10 pts) Derive the system of equations for salt content in two
MATH 528: Fall 2015
Due: 15 Sept 2015 (in class)
Homework 4
Numerical Methods for First-Order ODEs
You are encouraged (but not required) to use a computer to assist you with this problem.
Include a printout of any code you use to complete this problem.
HOMEWORK 2
Sec. 2.2 # 4 and # 22 Find the general solution to y + 4y + ( 2 + 4)y = 0 and the
solution to the IVP y(1/2) = 1, y (1/2) = 2.
Solution:
The characteristic equation is r2 + 4r + 2 + 4 = 0, with two roots given by the quadratic equation
p
p
4 42
PAIRS OF COMPLEX EIGENVALUES
We wish to find the general solution to
~y = A~y
(1)
Suppose the matrix A (which is nn with n 2) has a pair of complex conjugate eigenvalues
1,2 = i and complex conjugate eigenvectors ~1,2 = ~a ~bi. Then we know part of the
ge
HOMEWORK 5
Problem 1:
Find a general solution of the ODE
y 000 + 2y 00
y0
2y = 0
by first converting it to a system and as given.
Solution: Let u1 = y, u2 = y 0 , and u3 = y 00 . This transforms our single third order ODE into a system
of first order ODEs
HOMEWORK 5
Problem 1:
Find a general solution of the ODE
y 000 + 2y 00 y 0 2y = 0
by first converting it to a system and as given.
Solution: Let u1 = y, u2 = y 0 , and u3 = y 00 . This transforms our single third order ODE into a system
of first order ODE
MATH 528 - HOMEWORK 3
Problem 1:
(2.7.7) Find a (real) general solution.
3
9
(D2 + 2D + I)y = 3ex + x
4
2
Solution:
Step 1. General solution of the homogeneous ODE. The
ODE is
3
1
2 + 2 + = ( + )( +
4
2
characteristic equation of the homogeneous
3
)=0
2
w
EXAMPLE OF VARIATION OF PARAMETERS
We wish to find the general solution to
(1)
y + 4y = sin(2x)
We know the general solution to the homogeneous equation y + 4y = 0 is
(2)
yh (x) = c1 sin 2x + cos 2x
The Wronskian is
sin 2x
cos
2x
= 2
W =
2 cos 2x 2 sin
#2: The function |sin(x)| is clearly even and pi-periodic
Because it is even, we use a cosine series, starting with the constant term and then the coefficients of the non-constant cosine
pieces:
In[1]:=
1 Pi Integrate Abs Sin x
, x, 0, Pi
2
Out[1]=
In[2]:
MATH 528: Fall 2815 - Due: 8 Sept 2015 (in class)
Homework 3
a Method of Undetermined Coefcients
m Find a general solution of the following:
W
15-. y + 5y + 43; = 2 cosh 2:1: (Hint: rewrite cosh 2:6 as a sum of exponentials).
"N ,. in
32.52? y-l5y +4y=2co
PART 1.
Given: TA = 70
dT
From Eq (6) on p. 15:
= k(T TA )
dT
= k(T 70)
dt
dT
= kdt
T 70
ln|T 70| = kt + C1
eln|T 70| = ekt+C1
|T 70| = ekt eC1
T 70 = C2 ekt
T (t) = C2 ekt + 70
dt
Now let's use what we know to solve for C2 and k:
Let's call the 1st
Homework 2 Solutions
y = g(y) = y(T y)(K y)
0<T <K
PART 1a.
The equilibrium points can be found where g(y) = 0. It is easy to see that
g(y) = 0 at:
y = 0, T, K
To determine whether each is stable or unstable we need to look at the sign
of g (y) at the equ
August 7, 2012 21:04
c05
Sheet number 36 Page number 282
282
cyan black
Chapter 5. Series Solutions of Second Order Linear Equations
47. x2 y + xy + (x2 2 )y = 0,
48. y 2xy + y = 0,
49. y xy = 0,
Bessel equation
Hermite equation
Airy equation
5.5 Series S
MATH 528: Fall 2015
Due: 8 Sept 2015 (in class)
Homework 3
Method of Undetermined Coefficients
Find a general solution of the following:
1. y 00 + 5y 0 + 4y = 2 cosh 2x (Hint: rewrite cosh 2x as a sum of exponentials).
2. y 00 + 5y 0 + 4y = 2 cosh x.
3.
MATH 528: Fall 2015
Due: 1 Sept 2015 (in class)
Homework 2
All problems are from the textbook (Kreyszig, 10 ed.) unless otherwise specified. Only one
problem from each section will be graded, as selected by the grader.
Problem Set 2.1 - Reduction of Orde
MATH 528: Fall 2015
Due: 25 August 2015 (in class)
Homework 1
All problems are from the textbook (Kreyszig, 10 ed.) unless otherwise specified. Only one
problem from each section will be graded, as selected by the grader.
Problem Set 1.1 - Direct Integra
MATH 528: Fall 2015
Due: 22 September 2015 (in class)
Homework 5
All problems are from the textbook (Kreyszig, 10 ed.) unless otherwise specified. Only
certain problems from each section will be graded, as selected by the grader.
Constant-Coefficient Sys
EXAM 2 SOLUTIONS
#1: y + 4y = cos(2t)
(1a) Mathematica (with some helpful error checking):
(1b) Now we go to MATLAB:
We want to solve the ODE, plot the numerical result along with the analytically-obtained
solution from (1a), and report back the number of