Series of May/June 2013 past year questions: Part 1. This is a 9 mark question on differential equations in Alevels P3 May/June 2013 9709/33 question no 8.

The variables (x) and (t) satisfy the differential equation
[tdfrac{dx}{dt} = dfrac{k-x^3}{2x^2}]
for (t > 0), where (k) is a constant. When (t = 1), (x = 1) and when (t = 4), (x = 2).

(i) Solve the differential equation, finding the value of (k) and obtaining an expression of (x) in terms of (t). [9]

(ii) State what happens to the value of (x) as (t) becomes large. [1]