Use separation of variables to find the solution to the differential equation

Question

Use separation of variables to find the solution to the differential equation

3(du/dt) = u²

subject to the initial condition a(0) = 4.

u(t) = _____

Answer 1

I assume that you mean u(0) = 4.

3(du/dt) = u²

u^-2du = (1/3)dt

∫ u^-2du = (1/3) ∫dt

-1/u = (1/3)t + C

Since u = 4 when t = 0, C = -1/4.

-1/u = (1/3)t - 1/4

1/u = 1/4 - (1/3)t

u(t) = 1 / [1/4 - 1/3)t] = 1 / [(3 - 4t) / 12]

u(t) = 12 / (3 - 4t)