MATB44 Assignment 1 Question 3: solution
(3) Determine a model for the population P (t) if both the birth and the
death rate are proportional to the population present at time t.
Solution:
The birth rate is
P (t).
The death rate is
P (t).
The birth rate i

Mathematics MATB44, Assignment 4, November 29, 2011
Solutions to Selected Problems
1. (7.4 # 6)
(a) The Wronskian of the two solutions is
t t2
1 2t
det
= t2 .
(b) The Wronskian vanishes at t = 0. Hence the two vectors are linearly
independent on t < 0 and

MAT B44 Midterm Exam , Wednesday, November 2, 2011
5 pm 7 pm
No books or calculators are allowed.
SOLUTION
1. (20 points) Suppose the birth rate (births per unit time per unit
population) is proportional to the population, while the death rate (deaths
per

MATB44 November 15 and 17
Jordan Bell
Department of Mathematics, University of Toronto
November 28, 2016
System of linear first order ODE
x01 (t) = p1,1 (t)x1 (t) + + p1,n (t)xn (t) + g1 (t)
.
.
x0n (t) = pn,1 (t)x1 (t) + + pn,n (t)xn (t) + gn (t).
This c

MATB44 November 8 and 10
Jordan Bell
Department of Mathematics, University of Toronto
November 15, 2016
Regular singular points For y 00 + py 0 + qy = 0, x0 is a regular singular
point if p(x) and q(x) are defined at x = x0 but (x x0 )p(x) and (x x0 )2 q(

MATB44 November 22 and 24
Jordan Bell
Department of Mathematics, University of Toronto
December 3, 2016
Let A be a square matrix. Define
X
An
.
exp(A) =
n!
n=0
The term n = 0 is defined by
A0
0!
= I, the identity matrix. Let (t) = exp(At).
d X (At)n
(t)

28.12.2011
THERMODYNAMICS
(TER 201 E , 2011-2012 Fall, CRN: 13888)
2nd Midterm Exam
1) A 0.2 m3 rigid tank equipped with a pressure regulator contains steam(superheated water
vapour) at 2 MPa and 300C. The steam in the tank is heated. The regulator keeps

Denition: A function F : A Rn Rn is called a vector eld.
F (x) = F1(x), F2(x), , Fn(x)
Sometimes a function f : Rn R is called a scalar eld. The
components of a vector eld are scalar elds.
Denition: If F is a vector eld, a ow line for F is a path (t)
such

Theorem: Suppose f : Rn R is of class C 1 (smooth) and
: [a, b] Rn is a piecewise C 1 (smooth) path in Rn then
f ds = f (b) f (a) .
Corollary (1): If F =
f is a conservative vector eld and is a
curve joining x0 and x1, then
F ds = f (x1) f (x0).
Remarks:

Denition: A function f (x) dened for a x b is piecewise continuous in [a, b] if there is a nite partition, a = t0 <
t1 < < tn = b, such that f (x) is continuous for x (ti1, ti),
i = 1, , n and lim f (x) and lim f (x) exist and are nite.
xt+ 1
i
xt
i
Note:

The total square error of a function g (x) relative to f (x) is
dened to be the integral
2
f (x) g (x) dx .
E=
If f (x) is piecewise continuous on [, ], we get
12
a+
20
N
(a2
k
k =1
+
b2 )
k
1
[ f (x) ]2 dx =< f, f >
(a2 + b2 )
k
k
(called Bessels Inequal

Let
: b]
[a,
R
Rn .
Recall (from B41) that
1 (t)
2 (t)
3 (t) provided each (t) exists, 1 i n. We
D (t) =
i
.
.
.
n(t)
will write D (t) as (t) = 1 (t), 2 (t), , n(t) , a vector
in Rn.
If (t) exists, is called a dierentiable path.
If is a dieren

Denition: Let F be a vector eld in Rn and let : [a, b] Rn be
a smooth curve in Rn. The line integral of f over , denoted
F ds, is dened by
F ds =
(F T ) ds .
Denition: A (rst order) dierential form (1form) in Rn
is an expression of the form
f1 dx1 + f2 dx

MATB44 November 29
Jordan Bell
Department of Mathematics, University of Toronto
December 5, 2016
An ODE
x0 (t) = P (t)x(t) + g(t)
is called nonhomogeneous when g = 0. To solve this we use variation of
parameters.
To solve a nonhomogeneous ODE we first sol