Method of isoclines

Question

I have this exercise and I do not know how to solve it.

By using the method of isoclines represent the integrals of equation corbes nonautonomous $x'=x^2-t$.

There are some indications: Let $P = I_0$, the parabola $x^2 = t$. Show that $P^-$ is a trapping region ($P = P^+\cup P^-$).

How can we prove it?

Answer 1

The curve $t=x^2$ divides the $t$-$x$ plane into three distinct regions:

1. on the parabola, $t=x^2$ so $x'(t)=0$ and thus solutions of the DE are flat
2. inside the parabola, $x^2<t$ so $x'(t)<0$ and thus solutions are decreasing in $t$
3. outside the parabola, $x^2>t$ so $x'(t)>0$ and thus solutions are increasing in $t$

as this plot demonstrates: