Method 1:
2 du / dt = u^2
2du = u^2 dt (Only divide by u^2 leaving the 2 alone) -->
2/u^2 du = dt
2u^-2 du = dt
2*(u^-1)/-1 --> -2/ u = t + C (Integrate)
u(t + C) = -2 --> u = -2 / (t + C)
1 = -2 / ((0) + C)
1 = -2 / C
C = -2
u = -2 / (t - 2)
u = 2 / -(t - 2)
u = 2 / (-t + 2)
u = 2 / (2 - t).
Method 2:
du/u^2 = 1/2 dt (this time we divided by 2 AND u^2)
u^-2 du = 1/2 dt
-1/u = (1/2t) + C
u = -1/((1/2t) + C)
1 = -1/((1/2(0)) + C)
1 = -1/(C)
C = -1
u = -1/(1/2t - 1).
I would try to rearrange and see how "u" could be solved for faster depending on the simplicity of the integration...also the 2nd method has the simplest form without even doing anything...
The "nows" will add up to a later depending on the time restriction and how many actions took place...math is wonderful with this type of material!!!