Recently I've encountered the following equation in a book (don't remember the source, I just took some notes down on my scrap paper)
$\dot x=t-x^2,\quad x(1)=1.$ Then I've recognized that this is a Riccati type equation, but not explicitly solvable. The exercise proposed there was however to make a qualitative study of the equation.
In particular (and I am not able to solve any of the points, my bad):
show that the solution to the Cauchy Problem is defined on $[1,+\infty)$,
show that $\lim_{t\to+\infty}x(t)=+\infty,$
show that $\lim_{t\to+\infty}x(t)-t^{1/2}=0.$
Can anybody help me please? It is not homework. Thank you in advance
Guido