Lecture 09
Lecture 10
Lecture 11
Lecture 13 (exam in 12)
Lecture 14
Lecture 15
Added material to 13-15: chemostat example
Lecture 16
Lecture 17
A discrete one-species model (Sec 32)
N(t)
T
E t
t0 t1
N(t) := # individuals at time t at discrete times ti
wit

F2015 Math 321 Assignment 6 Solutions
34.3 N (t) = N0 eR(tt0 ) . We know N (t0 ) = 1500 and N (t0 +1) = 2000. Hence, 2000 = 1500eR ,
so eR = 4/3. Then N (t) = N0 (eR )tt0 = 1500(4/3)tt0 , so N (t0 + 4) = 1500(4/3)4 .
34.7 With an instantaneous growth rate

F2015 Math 321 Assignment 5 Solutions
28.1 (a) Using Taylor series, we have
sin = sin + ( ) cos + ( ).
(b) Let v = d/dt. Then using equation (28.2),
dv
g( ) kv
g kv
=
.=
,
d
Lv
Lv
= .
(c) The line v = will be a solution curve if
g k
g k
,
=
=
L
L
i.e., i

F2015 Math 321 Assignment 4 Solutions
22.1 (a),(b) Multiply the equation md2 x/dt2 + kx = Ff by dx/dt and rewrite in the form
d m
dt 2
dx
dt
2
k
+ x2 Ff x = 0.
2
Integrating the above, we get
2
m dx
k
+ x2 Ff x = E,
2 dt
2
Completing the square, we get
m

F2015 Math 321 Assignment 3 Solutions
18.3 For the equation md2 x/dt2 = f (x), the equilibrium positions are the values of x for
which f (x) = 0. Here f (x) = x2 x, so the equilibrium positions are x = 0 and x = 1. If
f (xE ) > 0, then the equilibrium pos

F2015 Math 321 Assignment 2 Solutions
8.2 We have seen that the general solution of the dierential equation md2 x/dt2 = kx
is given by x = c1 cos(t) + c2 sin(t), where = k/m. If we take the initial conditions
x(0) = x0 and (dx/dt)(0) = v0 , then we nd tha

F2015 Math 321 Assignment 1 Solutions
4.2 Using Newtons law to describe the motion in the horizontal and vertical directions,
we have
d2 x
d2 y
= 0,
= g.
dt2
dt2
Assuming that at time t = 0 the mass is at the end of the table (we denote this position by
x

Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22: exam
Lecture 23
Lecture 24
Introduction to Trac Flow (Sec 56)
Trac as uid
helicopter view of trac: cars moving at various speeds:
on some stretches light and fast, on others heavy and slow
one can ob

Extra problem for problems et 9 (with thanks to the student who suggested it!)
This problem is just to test the understanding of the technique we used in class to prove
that the Lotka-Volterra system has closed orbits.
Suppose that we want to plot the fol