1. A person is trying to fill her new swimming pool for the first time. Water is flowing into the pool at a constant rate of 300 litres/min. Unfortunately there is a hole and a crack in the pool from which water is leaking. Water is leaking out of the hole at a rate proportional to the square of the amount of water currently in the pool while water is leaking out of the crack at a rate proportional to the amount of water currently in the pool. As it had rained the previous day, the pool already contained 200 litres of water before filling began. Let t (mins.) be the time since the person started filling the pool and let W(t) be the number of litres of water in the pool at time t. Write down, but do not solve, the differential equation for W(t) along with its initial condition.
2. Solve the logistic equation
dP/dt = 3P (4-P) with P(0) = 2
What value does P approach as t gets large, ie. as t →∞.
3. (a) Find the general solution of
d2y/dt2 + d dy/dt - 4y = 0.
d2y/dt2 + d dy/dt - 4y = 8cos2t + 6sin2t
with y(0) = 4, y(0) = 0.
4. Solve the following differential equations:
a) (x2 + 4)dy/dx = xy.
b) d2y/dx2 + 4 dy/dx + 20y = 0.
c) dy/dx = y.
d) dy/dx + y = x2 - 2.
e) y" + 100y= 0.
f) x2 dy/dx + xy = x2sinx.
5. A dice is found to be weighted so that the chance of throwing a 6 is four times the chance of throwing a 1. Chances of throwing numbers other than 6 are equally likely.
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