Let $x=x(t)$ and for a constant $a\in (0,1)$ consider the initial value problem $$ x'(t) = x (1-x)(x-a),\quad 0\leq x(0)<a. $$
I want to show that this implies that $x(t)\downarrow 0$ as $t\to\infty$.
Do I have to solve by separating variables, i.e. $$ \int\frac{1}{x(1-x)(x-a)}\, dx= t +c? $$