University of Waterloo
AMath 350
Midterm Examination
Winter 2013
Friday March 1st , 6:30-8:00pm (90 minutes)
Name (print):
I.D. Number:
Signature:
Closed book.
Calculators NOT permitted.
MARKS
Question
Marks Available
1
10
2
14
3
10
4
11
5
12
6
13
Total
7

AMath 350
Assignment #3
Winter 2013
Solutions
1. (Find all solutions to the following equations:)
a) x2
dy
= xy
dx
y2
dy
y y2
=
, and we
dx
x x2
y
can see that it is a so-called homogeneous equation. So, we let u = ,
x
and then, in the usual way, we can w

AMath 350
Assignment #2
Fall 2011
SOLUTIONS
1. (Are the given equations separable, linear, neither, or both?)
a) Not separable, but linear.
b) Separable, but non linear.
c) Both.
d) Neither.
e) Both.
f) Separable, but not linear.
2.
a) Find the general so

University of Waterloo
AMath 350
Midterm Examination
Wednesday, October 24th .
Fall 2012
Duration: 90 minutes
Name (print):
I.D. Number:
Signature:
Closed book.
Calculators NOT permitted.
MARKS
Question
Marks Available
1
12
2
12
3
15
4
11
5
10
6
5
Total
6

AMath 350
Assignment #8
Fall 2012
Solutions
1. Following the hint, we have
Z1
Z
(x x2 ) dx =
e
0
1
0
=e
Now, letting u = x
1
2
(b) Di erentiating
2
e (x x) dx =
1
4
Z
1
e (x
Z
1
h
12
2
x
e(
0
)
1
4
i
dx
12
2
) dx.
0
, this becomes
e
1
4
Z
1
2
e
u2
du
1
2

AMath 350
Assignment #7
Fall 2012
Solutions
1.
a)
ax2
d2 y
dy
+ bx + cy = 0
2
dx
dx
Let x = ez (which, of course, means z = ln x, if x > 0). Then
dy
dy dz
1 dy
=
=
.
dx
dz dx
x dz
We also need the second derivative:
d2 y
d dy
d 1 dy
=
=
dx2
dx dx
dx x dz

AMath 350
Assignment #9
Fall 2012
Solutions
1. Determine if the transforms exist:
a) f (x) = x
Z1
Z1
i! x
xe
dx converges ()
xe
1
0
verge.
i! x
dx and
Z
0
xe
i! x
dx both con-
1
t
xe i!x e i!x
xe
dx 350
= lim
xe
dx = lim 7 Solutions 2
+
t!1 0
t!1
i!
!
0
0

AMath 350
Assignment #6
Fall 2012
Solutions
1.
a)
y 00 + ky = 0,
y (0) = 0,
y (1) = 1
Since the characteristic equation is m2 + k = 0, so m =
consider three cases.
p
k , we must
Case I: k < 0
If k < 0, then y = C1 e
require that
p
kx
p
+ C2 e
kx
. The gi

AMath 350
Assignment #7
Winter 2013
Due Friday, March 15th
Hand in #3, #5, #7, and #9
Note: The pace of the assignments has fallen behind the lectures a bit. This
assignment will get us caught up, which is why its a bit longer than usual. It covers
inhomo

AMath 350
Assignment #9
Winter 2013
Due Friday, April 5th
Hand in #3, #4, #5, and #6ab.
1. Use the denition or Theorem 5.31 of the Course Notes to determine if the
Fourier Transform of each of the following functions exists.
a) f (x) = x
2
b) f (x) = xe x

AMath 350
Assignment #8
Winter 2013
Due Monday, March 25th
Hand in #1, #3, #5, #6, and #7
1. This rst question involves just a bit more practice with denite integrals, as
preparation for some work to come in class.
The error function is dened as
2
erf (x)

AMath 350
Assignment #6
Winter 2013
Due Friday, March 8th
1. Earlier in this course we saw that a variety of 1st-order DEs can be solved
with the help of substitutions. Substitutions can help with some 2nd-order
equations as well. An equation of the form

AMath 350
Assignment #4
Winter 2013
due Friday, February 8th
Hand in #1, #2c, #4, #5, and #6.
1. Find the general solutions to the following ODEs:
a) y 00
5y 0 + 6y = 0
b) y 00 + 4y 0 + 4y = 0
c) y 00 + 4y 0 + (4 + 2 ) y = 0
d) y 000
3y 00 + 3y 0
y=0
e) y

AMath 350
Assignment #2
Winter 2013
due Wednesday, January 23rd
Hand in #2, 5, and 6.
1. Are the following equations separable, linear, neither, or both?
a)
b)
c)
d)
e)
f)
2.
dy
= y + ex
dx
dy
= ex sin( y )
dx
dy
= 5y 2
dx
dy
= x y2
dx
dy
x2
=y
dx
dy
y
=x

AMath 350
Assignment #1 (Review)
Winter 2013
Not for Submission
1. Calculate the following indenite integrals:
Z
a)
y sin (y ) dy
Z
1
b)
dt
4 4t + t2
Z
ln (2x)
c)
dx
x
Z
d)
sin2 (3) d
Z
1
e)
du
9 u2
Z
1
p
f)
dx
1 x2
Z
g)
tan () d
2. Determine if the follo

AMath 350
Assignment #3
Winter 2013
due Wednesday, January 30th
Hand in #1, #2b, #4, and #6a
1. Find all solutions to the following equations, by making an appropriate substitution:
dy
= xy y 2
dx
dy
b)
y = e 2x y 3
dx
dy
c)
= (4x + y )2 ,
dx
a) x2
subjec

AMath 350
Assignment #4
Winter 2013
Solutions
1.
a) y 00 + y 0 2y = 0
Solution: The characteristic equation is m2 + m 2 = 0, which gives
(m + 2)(m 1) = 0 =) m = 2, 1. Therefore the general solution is
y = c1 e
2x
+ c2 e x .
b) y 00 + 4y 0 + 4y = 0
Solutio

AMath 350
Assignment #5
Winter 2013
Solutions
1. Solve y 0 + 3y = 5e
3x
+ 6xe3x :
By inspection, the complementary function is yh = Ce
3x
.
For a particular solution, the 5e 3x term would normally require a term of the
form Ae 3x , but since this matches

University of Waterloo
AMath 350
Midterm Examination Fall 2012
Wednesday, October 24. Duration: 90 minutes
Name (print):
I. *RQLOX H 1') \Q .5
LB. Number: _,_,_._._._+._h
\
Signature:
Closed book.
Calculators 1M permitted.
MARKS
Marks Available Marks