In many models of oscillations in science, the population of one oscillating species (p) can be modelled as p(t) = p0e λt , where λ is a solution of λ
2 + βλ + α
2 = 0, for some constants α and β, which can represent a wide range of things, depending on the scientific context. They could be related to the birth rate of a prey species, for example, or to the rate at which an ion is removed from a cell, or to the action of a hormone on a neuron in the pituitary gland, etc

a. Show that, when β = 0, λ = ±|α|i.

b. Show that λ is complex when |β| < 2|α| (Hint: use the formula for the roots of a quadratic.)

c. When λ = αi (i.e., when β = 0) what is the solution p?
(Hint: write p(t) = x(t)+iy(t) and then work out what x(t)
and y(t) are.)

d. Now suppose that 0 < β < 2|α|. We know that λ will now be a complex number with a non-zero real part. How does this change the real part of the solution p(t)? (Hint: first write λ as a complex number, i.e., λ = γ + ωi, and work out what γ and ω are. Then put this expression for λ into the equation for p(t), to work out what the solution p(t)
will look like.)