Questão 1: (2 pontos)

Consider the following separable differential equation

(a) Write this equation in the form .

Enter your answer in the form .

__________

(b) Integrate both sides, to get a solution of the form .

Enter your answer in the form .

__________

cos2 x y′ = 5y2

∫ P(y) dy = ∫ Q(x) dx

(P(y),Q(x))

R(y) = S(x) + constant

(R(y),S(x))

Questão 2: (3 pontos)

Consider the following linear differential equation,

(a) Calculate the Integrating factor.

__________

(b) Hence calculate the solution.

Note: You must use lowercase c for your constant of integration.

__________

(c) Given the initial condition , calculate the value of the constant of integration c.

Note: You must enter your answer exactly, not as a decimal.

Questão 3: (2 pontos)

Consider the following initial value problem

,

(a) Find .

____________

(b) Find .

____________

Questão 4: (7 pontos)

Consider a population whose number we denote by . Suppose that is the average number of births per capita per year and is

the average number of deaths per capita per year. Then the rate of change of the population is given by the following differential

equation,

where is the time (in years).

(a) Suppose that and . Solve the above ODE subject to the initial condition , and enter your expression for

below:

__________

(b) For your solution above, what happens as ?

__________

By considering your solution, or the original ODE, if was bigger than , then as ,

__________

Note: is entered as “infinity”.

(c) If the following fraction is to be split using partial fractions,

then ____________ and ____________.

(d) Now suppose that the death rate is composed of a per capita death rate as before, plus a death rate due to overcrowding and

competition for resources of the form (it gets worse the bigger the population is), so that

Substitute this into the population differential equation (1) (with and as before) and solve for .

Hint: you might find the partial fraction in part (c) useful in determining your solution.

Enter your expression for below:

__________

(e) By examining your solution above, or by considering the differential equation for the population with the modified death rate (part

(d)), as ,

__________

= bP − dP, (1)

dP

dt

t

b = 6 d = 2 P(0) = 100

P(t)

P(t) =

t⟶∞

P ⟶

d b t⟶∞

P ⟶

∞

= + ,

1, 000

P(4, 000 − P)

A

P

B

4, 000 − P

A = B =

d

γP

d = 2 +

P

1, 000

b = 6 P(0) = 100 P(t)

P(t)

P(t) =

P

t⟶∞

P ⟶

4/28/2019 The University of Adelaide –