Mathematical modelling in the sciences
(ACM10070)
Assignment 4
Dr Lennon Ó Náraigh
Issued: Wednesday 5th October 2016
Due: Wednesday 12th October 2016 (in class)
Instructions: Even-numbered assignments are graded.
1. (a) Let P (t) be the average concentration of a pollutant in a particular domain
at time t. The pollutant naturally degrades over time at a rate k but the
domain is subject to a pollution source, so that pollutant enters the domain
at a constant (positive) rate s. These effects are summarised in the following
ordinary differential equation (ODE):
dP
= −k(P − P0 ) + s,
dt
k, s ∈ R+ ,
(1)
where P0 is the background level of pollution. Let P (0) = P0 be the initial
pollution level. Using the integrating-factor technique, show that
P (t) =
s
1 − e−kt + P0 .
k
(2)
(No marks if the answer is obtained by direct substitution of the proposed
solution (2) into the ODE itself.)
(b) Show that
s
lim P (t) = P0 + ,
t→∞
k
independent of the initial level of pollution. Hence, make a rough sketch of
the solution P (t) coming from Part (a).
(c) A powerplant emits nitrous oxides (NOx) at a rate s according to Equation (1). The factory is fined by the Environmental Protection Agency if the
pollution level (even momentarily) exceeds twice the background level, i.e. a
fine is imposed if P (t) > 2P0 . Show that the factory should emit at a rate
s < P0 k
to avoid the fine.
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Mathematical modelling in the sciences
Qualitative methods for ODEs
2. Sketch the one-dimensional vector field for the following autonomous ODEs:
dy
= y cos(y),
dt
dy
= y(y − 1) 1 − 12 y .
dt
3. (a) Using separation of variables, show that the solution to the ODE
dz
= kz,
dt
z(0) = z0
is z(t) = z0 ekt . Here k is a constant value.
(b) Consider a generic first-order autonomous ODE,
dy
= f (y).
dt
(3)
Let y(t) = y∗ + δ(t), where y∗ is one of the fixed points of Equation (3) and
where δ(t) is a small quantity. Using the best-straight-line-approximation,
show that
d
δ = f 0 (y∗ )δ,
δ(0) = δ0 ,
|δ0 | 1.
dt
0
Hence, show that δ(t) = δ0 e[f (y∗ )]t .
(c) Consider again the second ODE in Question 2, which we rewrite here as
dy
= f (y),
dt
f (y) = y(y − 1) 1 − 21 y .
(4)
For each of the different fixed points y∗ , comment on the following:
• The sign of f 0 (y∗ );
• Whether δ(t) is exponentially increasing or exponentially decreasing;
• How the sign of f 0 (y∗ ) relates to the stability or otherwise of the fixed
point y∗ as determined from the one-dimensional vector field.
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