Understanding qualitative theory and stability

I have this question in an exercise book, I have the answer but I do not understand how the answer is derived, can anyone help with a working solution...

differential equation....

I know the equation is separable, but how can I draw the integral curves (tx - plane)

what happens to the the solution as t reaches infinity...

I'm also asked to draw the two equilibrium states and decide whether they are stable or not

and for the second part i have a solution but not sure if it is correct or now...

any help would be great thanks.