MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 17
Due Friday March 27th, 2009
(1) For each of the following ODEs, determine whether x = 0 is an ordinary or singular point.
If it is singular, determine whether it is regular or not. (Remember, rst
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 18
Due Monday March 30th, 2009
(1) For each of the following sets, determine if it is a vector space. If it is, give a basis. If it
isnt, explain why not.
(a) The set of points in R3 with x = 0.
(b)
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 19
Due Friday April 3rd, 2009
(1) The function
t < /2,
0
1
/2 t < /2,
f (t) =
0
/2 t ,
can be extended to be periodic of period 2 . Sketch the graph of the resulting function,
and compute its Fouri
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 20
Due Monday April 6th, 2009
(1) Determine which of the following functions are even, which are odd, and which are neither
even nor odd:
(a) f (t) = t3 + 3t.
(b) f (t) = t2 + t.
(c) f (t) = et .
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 21
Due Friday April 10th, 2009
(1) Compute the complex Fourier series for the function dened on the interval [, ]:
f (x) =
1,
4,
x < 0,
0 x .
Use the cn s to nd the coecients of the real Fourier se
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 22
Due Wednesday April 15th, 2009
(1) Let X be a vector space over C (i.e., the contants are complex numbers, instead of just
real numbers). If cfw_v1 , v2 is a basis of X , then by denition, every
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 23
Due Monday April 20th, 2009
(1) Let u(x, t) be the temperature of a bar of length 10, at position x and time t (in hours).
Suppose that initially, the temperature increases linearly from 70 at th
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 24
Due Friday April 24th, 2009
(1) Which of the following functions are harmonic?
(a) f (x) = 10 3x.
(b) f (x, y ) = x2 + y 2 .
(c) f (x, y ) = x2 y 2 .
(d) f (x, y ) = ex cos y .
(e) f (x, y ) = x3
MTHSC 208, Introduction to Ordinary Dierential Equations, Sec. 10
Spring Semester 2009
MWF 2:303:20, Th 2:002:50
M102 Martin Hall
General information
Instructor:
Email:
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Dr. Matthew Macauley
[email protected]
htt
72b 30
FALL 2011 MTHSC 208
NAME:
TESTZ P1
1. Find and plot both the steady periodic solution ysp = Ccos(wt ~ 0:) and the transient
somtion yrr that satisfies the given iVP
y+8y'+25y = 200 cost + 520 sin z; y(0) : 30,y'(0) = 10
. 2. .4.
. W; i.
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 16
Due Monday March 23rd, 2009
(1) Use the ratio test to nd the radius of convergence of the following power series:
1
(x )n ,
n+1
n=0
b.
n=0
d.
(1)n xn ,
a.
1
(x )n ,
2n
n=0
c.
(5x 10)n ,
e.
3
(x 2
MTHSC 208 (Dierential Equations)
Dr. Matthew Macauley
HW 15
Due Monday March 9th, 2009
(1) Find the general solution of x2 y xy 3y = 0.
(2) Find the general solution of x2 y xy + 5y = 0.
(3) Find the general solution of x2 y 3xy + 4y = 0.
(4) Consider the
MthSc 208 (Spring 2011)
Worksheet 5a
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 5a: Laplace Transforms
NAME:
The Laplace transform of a function f (t) is the function F (s) := Lcfw_f (t)(s) =
0
f (t)est dt.
1. Compute the Laplace
MthSc 208 (Spring 2011)
Worksheet 5b
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 5b: Properties of Laplace Transforms
NAME:
Consider the following properties of the Laplace transform: (i) Lcfw_ect f (t)(s) = F (s  c) (ii) Lcfw_tn f
MthSc 208 (Spring 2011)
Worksheet 5c
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 5c: Solving ODEs with Laplace Transforms
NAME:
Consider the initial value problem: y  y = e2t , y(0) = 0, y (0) = 1. The following facts will be usefu
MthSc 208 (Spring 2011)
Worksheet 5e
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 5e: Laplace Transforms and the Heavyside Function
NAME:
Recall the following properties of the Laplace transform: (i) Lcfw_eat (s) = (ii) Lcfw_tn (s) =
MthSc 208 (Spring 2011)
Worksheet 5f
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 5f: ODEs with Piecewise Forcing Terms
NAME:
Consider the initial value problem y + y = f (t), y(0) = 0, y (0) = 1, where f (t) =
2t, 2,
0t1 t>1
1. Sket
MthSc 208 (Spring 2011)
Worksheet 6a
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 6a: Fourier Series
NAME:
Consider the square wave defined by f (x) =
1, 1,
0x< and extended to be 2periodic.  x < 0
1. Sketch f (x) and find its Fou
MthSc 208 (Spring 2011)
Worksheet 6b
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 6b: Complex Fourier Series
NAME:
Consider the square wave defined by f (x) =
1, 1,
0x< and extended to be 2periodic.  x < 0
1. Sketch f (x) and find
MthSc 208 (Spring 2011)
Worksheet 6c
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 6c: Parseval's Identity
NAME:
Recall that Parseval's identity says that 1
f (x)

2
1 dx = a2 + 2 0
(a2 + b2 ) . n n
n=1
We will use this to compute
MthSc 208 (Spring 2011)
Worksheet 7a
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 7a: The Heat Equation
NAME:
We will solve for the function u(x, t) defined for 0 x and t 0 which satisfies the following initial value problem of the h
MthSc 208 (Spring 2011)
Worksheet 7b
MthSc 208: Differential Equations (Spring 2011) Inclass Worksheet 7b: The Wave Equation
NAME:
We will solve for the function u(x, t) defined for 0 x and t 0 which satisfies the following initial value problem of the w