Solution
= Solution
Take
$$
F=x^2+y^2.
$$
Then
$$
\dot F=2x\dot x+2y\dot y
=2k(x^2+y^2)^2
=2kF^2.
$$
With $F(0)=1$, separation gives
$$
-\frac1F=2kt-1,
$$
and hence
$$
\boxed{F(t)=\frac1{1-2kt}}
$$
on its maximal interval of existence.
= Solution
Take
$$
F=x^2+y^2.
$$
Then
$$
\dot F=2x\dot x+2y\dot y
=2k(x^2+y^2)^2
=2kF^2.
$$
With $F(0)=1$, separation gives
$$
-\frac1F=2kt-1,
$$
and hence
$$
\boxed{F(t)=\frac1{1-2kt}}
$$
on its maximal interval of existence.