question 1 a the time rate of change of a falling object s height can

Question

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.