MATHEMATICS 2C3 HOMEWORK WINTER 2012
ERIC T. SAWYER
Week #1: January 4,5
(1) (3.1 in Z&C) A thermometer is taken from inside a building at 20 C and
placed outside where the temperature is 10 C . After half a minute the
thermometer reads 10 C . How much lo
Math 2C3 Assignment #7 Solutions
March 14, 2012
1. (6.1 #2, #4 page 230 Z&C): Find the radius of convergence and interval of convergence
for each of the series
(a)
P1
n=0
100n
n!
(x + 7)n ;
Solution: Compute the limit of the ratio of moduli of successive
Math 2C3 Assignment #3 Solutions
February 2, 2012
1. (3.2 #2 page 99 Z&C): The number N (t) of people in a community who are exposed
to a particular advertisement is governed by the logisitic equation
dN
= N (a bN ) :
dt
Initially, N (0) = 500, and it is
Math 2C3 Assignment #3
Assignment due 1:30pm Wednesday February 1 in HH locker
January 25, 2012
1. (3.2 #2 page 99 Z&C): The number N (t) of people in a community who are exposed
to a paricular advertisement is governed by the logisitic equation
dN
= N (a
Math 2C3 Assignment #4 Solutions
February 7, 2012
1. (4.1 #18 page 129 Z&C): Determine whether or not the set of functions
cos 2x; 1; cos2 x
is linearly independent on the interval ( 1; 1).
Solution: The trigonometric formula cos 2x = 2 cos2 x 1 shows tha
Math 2C3 Assignment #5 Solutions
February 29, 2012
1. (4.3 #4 page 138 Z&C): Find the general solution of the equation
y 00
3y 0 + 2y = 0:
Solution: The characteristic polynomial is
P (r) = r2
3r + 2 = (r
1) (r
2) ;
and since the roots of P (r) are 1 and
Math 2C3 Assignment #6 Solutions
March 6, 2012
1. (4.4 #44 page 149 Z&C): Use the method of undetermined coe cients to nd a
particular solution to
y 00 + y = sin x cos 2x:
Solution: The characteristic polynomial is r2 + 1 with roots
mental solution set. T
Math 2C3 Assignment #6
Assignment due 1:30pm Wednesday March 7 in HH locker
March 1, 2012
1. (4.4 #44 page 149 Z&C): Use the method of undetermined coe cients to nd a
particular solution to
y 00 + y = sin x cos 2x:
2. (4.6 #26 page 162 Z&C): Use the metho
Math 2C3 Assignment #1
Assignment due 1:30pm Wednesday January 18 in HH locker
January 11, 2012
1. (1.1 #22 page 10 Z&C): Verify that the one parameter family of functions
Z x
2
2
x2
et dt + Ce x
y=e
0
solve the dierential equation
y 0 + 2xy = 1:
2. (1.2
Differential Equations
Differential equations describe relationships involving the derivatives of functions. For our
purposes, we will make two basic distinctions.
Definition: Pure-Time Differential Equation
A pure-time differential equation is a differen
Math 2C3 Assignment #2 Solutions
January 24, 2012
1. (2.1 #4 page 42 Z&C): The direction eld for the equation y 0 = sin x cos y is v =
p
1
;
1+sin2 x cos2 y
p
sin x cos y
1+sin2 x cos2 y
:
5
y
4
3
2
1
-5
-4
-3
-2
-1
1
2
-1
3
4
5
x
-2
-3
-4
-5
Direction el
Math 2C3 Assignment #1 Solutions
January 17, 2012
1. (1.1 #22 page 10 Z&C): Verify that the one parameter family of functions
Zx
2
2
x2
et dt + Ce x
y=e
0
solve the dierential equation
y 0 + 2xy = 1:
Solution: We calculate
d
y=
dx
0
e
x2
Z
x
t2
e dt + C
=
Math 2C3 Assignment #7
Assignment due 1:30pm Wednesday March 14 in HH locker
March 7, 2012
1. (6.1 #2, #4 page 230 Z&C): Find the radius of convergence and interval of convergence
for each of the series
(a)
(b)
P1
n=0
P1
n=0
100n
n!
(x + 7)n ;
n! (x
1)n :
Math 2C3 Assignment #8 Solutions
March 20, 2012
1. (6.2 #32 page 240 Z&C): The point x = 0 is a regular singular point of the equation
1) y 00 + 3y 0
x (x
2y = 0:
Show that the indicial roots dier by an integer. Use the recurrence relation with the
larger
Math 2C3 Assignment #8
Assignment due 1:30pm Wednesday March 21 in HH locker
March 14, 2012
1. (6.2 #32 page 240 Z&C): The point x = 0 is a regular singular point of the equation
x (x
1) y 00 + 3y 0
2y = 0:
Show that the indicial roots dier by an integer.
Math 2C3 Assignment #9 Solutions
March 27, 2012
1. (7.2 #4, #16, #26 page 269-270 Z&C): Find both L
Solution: We expand
(
L
1
12
s3
2
s
1
s3
2
s
2
4
s2
=
)
4 s14 +
1
s6
1
(x) = L
= 4x
We write
L
1
s+1
s2 +2
=
s+1
s2 + 2
s
p
2
s 2 + ( 2)
+
(x) = L
1
s2
+
n
Math 2C3 Assignment #9
Assignment due 1:30pm Wednesday March 28 in HH locker
March 21, 2012
1. (7.2 #4, #16, #26 page 269-270 Z&C): Find both L
1
n
12
s3
2
s
o
(x) and L
1
s+1
s2 +2
(x).
2. (7.2 #38 page 270 Z&C): Use the Laplace transform to solve the in
Math 2C3 Assignment #10 Solutions
April 3, 2012
1
1. (7.4 #32 page 290 Z&C): Find L
n
1
s2 (s 1)
o
(x).
n
o
Rx
Solution: L 1 s 1 1 (x) = ex , and so using the formula L 1 F (s) (x) = 0 f (t) dt twice,
s
we get
Zx
1
1
L
(x) =
et dt = ex 1;
s (s 1)
Z0 x
1
L
Math 2C3 Assignment #10
Assignment not to be handed in
April 2, 2012
1. (7.4 #32 page 290 Z&C): Find L
1
n
1
s2 (s 1)
o
(x).
2. (7.4 #54 page 291 Z&C): Find the Laplace transform of the periodic function f (x) of
period 2 with
sin x for 0 x
f (x) =
:
0
fo
Lecture Notes on Ordinary Dierential Equations
Eric T. Sawyer
McMaster University, Hamilton, Ontario
E-mail address : [email protected]
URL: http:/www.math.mcmastern~sawyer
Abstract. These lecture notes constitute an elementary introduction to the
theory
Picard iteration
The Theorem on Existence and Uniqueness states that if both f (t, y ) and f /y are continuous in some
region around the point (t0 , y0 ) then there is a unique solution to the initial value problem
y = f (t, y )
y (t0 ) = y0
(1)
valid in
tan x =
sin x =
1
csc x
sin x
cos x
cos x =
1
sec x
tan x =
1
cot x
sin y = x
cos y = x
tan y = x
y = arcsin x
y = arccos x
y = arctan x
sin2 x + cos2 x = 1
sin(u v ) = sin u cos v cos u sin v
cos(u v ) = cos u cos v sin u sin v
sin(n ) = 0, n Z
sin
IMPORTANT MESSAGE: COUSE OUTLINE IS TENTATIVE
The instructor and university reserve the right to modify elements of the course during the
term. The university may change the dates and deadlines for any or all courses in extreme
circumstances (such as a st