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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?

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Please let me know if I have changed the meaning of your question during my edit. – Michael Albanese Jan 4 '13 at 8:45
no no thank you . – kiroro Jan 4 '13 at 8:55
please help me ,i need help – kiroro Jan 4 '13 at 10:53
@Jasper Loy: what does this question have to do with graded modules (dynamical systems seems to be a perfectly appropriate tag) and why did you remove the pleasantries? The former seems to be a mistake and the latter a rather impolite thing to do. – Martin Jan 4 '13 at 11:12
Hi, please have you any idea to solve this exercise ? – kiroro Jan 4 '13 at 13:39
up vote 0 down vote accepted

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:

Mathematica graphics

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