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 #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
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 #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 #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 #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 #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 #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 #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
=
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
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
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
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
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 #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 #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 #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 #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 #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 #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