Differential Equations Bounded

Question

What does it mean when it says the solutions become unbounded in finite time? How do solutions that look bounded look like. I don't really understand how this question relates back to the generalized equation r^2 = x^2 + y^2

Answer 1

Hint.

(a)

The equilibrium points are those such that $x' = y' = 0\;$

Considering the linearised approximation about $(0,0)$ we have

$$ x' = y\\ y' = -x $$

which gives the orbits

$$ (r^2)' = 0 \Rightarrow r = C_0 $$

because $\frac 12(r^2)' = x\frac{dx}{dt}+y\frac{dy}{dt} \;$ characterizing a center.

(b)

Multiplying respectively by $x, y\;$ and adding we have

(1) $$ x\frac{dx}{dt}+y\frac{dy}{dt} = (x^2+y^2)^2\Rightarrow \frac 12 (r^2)' = r^4 $$

(2) $$ x\frac{dx}{dt}+y\frac{dy}{dt} = -(x^2+y^2)^2\Rightarrow \frac 12 (r^2)' = -r^4 $$