I have the Differential equation x' = x^2 - tx + a(t), where a(t) is a continuous function defined for R such that for every t in R, 0< a(t) <= 1.

I have shown that x = 0 is a lower fence.

The question is as follows:

Given ( t(0), x(0) ) element of R^2 with 0 =< x(0) < t(0). Show that the max solution passing through ( t(0), x(0) ) is defined up to + infinity.

I don't even know how to begin so any help would be great.