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MATH1400: Modelling with differential equations (Spring 2022)

Examples 3

Section 1: to be covered in tutorials.

1. Sketch the slope field of the ODE

y\ = sin2 y

and hence the form of the solutions.

Hint: Begin by plotting thefunction sin2 y as afunction of y, and identify the constant solutions of y for which y\ = sin2 y = 0.

2. A nuclear reactor produces waste that contains the isotope 239Pu (Plutonium-239). After 15 years, the concentration of 239Pu in the waste has decreased by 0.043%. Estimate the half-life.

3. A fresh cup of coffee has a temperature of 80◦ C. The coffee is cool enough to drink once its temperature is 40◦ C. The ambient temperature is 20◦ C. After 2 minutes, the coffee has cooled by 10◦ C. Using Newton’s law of cooling, estimate the total time, after the coffee was first prepared, for the coffee to be ready to drink.

4. The population p(t) of a country with a constant net negative migration is modelled using = µp ( 1 − ) − M,

with growth rate µ, carrying capacity p∞ and net migration rate M, all positive constants. (a) Show that if p is a constant solution (so p is a constant and dp/dt = 0), then it satisfies

p2 − p∞p + λp = 0,

where λ = M/µp∞ . By solving this quadratic equation, determine any constant solutions, distinguishing between whether λ < 1/4 or λ > 1/4.

(b) Using the results of (a) to help, sketch the slope fields of the ODE for the case (i) λ < 1/4,

and (ii) λ > 1/4. Hence sketch the form of the possible solutions in the two cases. Give the conditions on λ and p(0) required to retain a long-term stable population.

5. The temperature of a pond is modelled using Newton’s law of cooling. To capture the day-night cycle, the ambient temperature is assumed to vary sinusoidally in time, A(t) = A0 sin(ωt), so

dθ

dt = −k(θ − A0 sin(ωt)),

with θ(0) = 0, and k a constant. By solving the ODE, show that the solution θ(t) approaches

A0 k

k2 + ω 2

as t → ∞ .

Hence describe briefly how θ∞ (t) relates to A(t).

Section 2: to be handed in

1. Sketch the slope field of the ODE

y\ = −xy .

Hence sketch its solutions.

Hint: Begin by showing the slope along each axis (x = 0 and y = 0). Then consider the slope along the curve y = 1/x, then 2/x etc.

Determine the general solution to the ODE and compare the result to your sketch.

2. A nurse is administering a dose of iodine-131 (131I) to a patient with hyperthyroidism. Let N(t) denote the total number of iodine-131 atoms that are present in the dose. A doctor has specified that, at the time the dose is administered (t = 0), the dose must supply

decays per second.

Given that iodine-131 has a half-life of 8.1days, calculate the number of iodine-131 atoms, N(0), that must be prepared in the dose to ensure that the initial decay rate above is provided at t = 0.

3. The population of a country is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is 20,000, estimate the number of people initially living in the country.

4. The number of cells in a tumour, n(t), can be modelled using the differential equation

where n0 , λ and α are positive constants.

(a) Sketch the slope field of the ODE and hence the solutions.

Hint: Begin by plotting the gradient along the n-axis (setting t = 0).

(b) Show that n(t) = n0 exp ( ‘1 − e −αt ) ) . What happens as t → ∞ ?

(c) At t = 0, a tumour is observed to contain 104 cells and to be increasing at a rate of 20% per unit time (i.e. dn/dt = 0.2 n at t = 0). Using the model above with α = 0.02, determine the size of the tumour as t → ∞ .

5. Hint: This problem is mathematically similar to Q5 of section 1.