Question 1

a) The time rate of change of a falling object's height can be given by the differential equation

where y = height of the object, a = acceleration and t = time.

An object is dropped from a height of 500m, with an acceleration due to gravity of

-9.8m/sec²,

(i)Express the height of the object as a function of time.

(ii) When will the object reach ground level?

b) The acceleration (a =) of a car can be shown to obey the following linear differential

equation = 0.05(250-v), where v is the velocity of the car.

(i) Express the velocity of the car as a function of time.

(ii) How long does it take the car to accelerate from rest to 200km/h?

c) Find the general solution of

d'r dr

ds² ds - 2r=0

d) Find the first four non-zero terms of the power series solution for the differential equation

y" + x²y = 0.