Question 1

(a)  Show that y₁(t) = t and y₂(t) = e^t are linearly independent solutions of the homogeneous second order linear DE (1 - t)y'' + ty' -y = 0.

(b)  Hence use the method of variation of parameters to determine a particular solution of the inhomogeneous second order DE (1 - t)y'' + ty' - y = (t - 1)²e-t.

Question 2

A uniform bar of mass m₁ hangs from two identical springs attached to the bar equidistant from its centre of gravity, and a mass m₂ is suspended from a third spring, having the same spring constant as the other two, which is attached to the underside of the bar at its centre of gravity, as shown below:

Then the upward displacement y₁ of the bar from its equilibrium position and the upward displacement y₂ of the suspended mass from its equilibrium position are governed by the system of linear differential equations

m₁y₁′′ + 3ky₁ - ky₂ = 0

m₂y₂′′ - ky₁ + ky₂ = 0

where k is the spring constant for all three springs, and we suppose units are chosen so that displacement is measured in metres.

Now assume that m₁ = 3, m₂ = 1 and k = 2; and that the following initial conditions apply:

y₁(0) = 0, y₁'(0) = 0, y₂(0) = 1,  y₂'(0) =  0.

(a)  Write down the system of first order ODEs corresponding to the above system of 2nd order IVPs. (Do not solve the system).

(b)  Use Laplace transforms to solve the system

(c)  Determine how high the equilibrium position for m2 needs to be to avoid the mass hitting the bench;

(d)  If the motion of the mass and the bar are periodic, find the period; if not, describe the motion.

Hints:

1. A polynomial of the form as⁴ + bs² + c can be factorised using the quadratic formula.

2. The period of a sum of two sinosoids of rational frequencies ω₁ and ω₂ has period given by max(2Π⁄ω₁ 2Π⁄ω₂).

3.  A sum of two sinosoids of frequencies ω₁ and ω₂ is periodic, if and only if the ratio ω₁⁄ω₂ is rational.

4.  For part (d) it is useful to plot the graphs of y₁  and y₂.