Question

consider the system:
dx/dt = x^2+y
dy/dt = x^2*y^2
Show that, for the solution (x(t),y(t)) with initial ocndition (x(0),y(0)) = (0,1), there is a time t* such that x(t)--> infinity as t--> t*. In other words the solutions blows up in finite time.

Answer 1

$y(0)=1\ and\ dy/dt = x^2*y^2>=0\implies y(t)\ge 1$,

$\implies$ $dx/dt \ge x^2+1,\implies$

$\frac{dx/dt}{x^2+1}\ge 1 \implies$

$(arctan(x(t)))'_t>=1,\implies$

$arctan(x(t))\ge x(0)+t=0+t=t,$

Therefore there are bounded increasing sequence of points

${t_n},\ (t_n<\pi/2), t_n\to t^*\le\pi/2:$

$arctan(x(t_n))\to \pi/2\implies$

$x(t_n)\to\infty$,when $t_n\to t^*.$