Problem: Let the Hamiltonian of a dynamical system is given by $~H=pq-q^2~,$ where $~q~$ is the generalized co-ordinate and $~p~$ is the generalized momentum. Then as $~t\to\infty~$
$1.~~~q\to\infty~,~~p\to\infty$
$2.~~~q\to0~,~~p\to0$
$3.~~~q\to\infty~,~~p\to0$
$4.~~~q\to0~,~~p\to\infty$
My work: We know that the time evolution of the system is uniquely defined by Hamilton's equations: $$\dfrac{dq}{dt}=\dfrac{\partial H}{\partial p}\qquad\text{and}\qquad-\dfrac{dp}{dt}=\dfrac{\partial H}{\partial q}$$ $$\implies \dfrac{dq}{dt}=q\qquad\text{and}\qquad\dfrac{dp}{dt}=-p+2q$$Integrating, $$q=Ae^t\qquad\text{and}\qquad p=Ae^{2t}+Be^{-t}$$where $~A~,~B~$ are arbitrary constants.
Now when $~t\to\infty~,$ then $~~~q\to\infty~,~~p\to\infty~.$
Hence only option $(1)$ is true.
My query: But given that only correct option is option $(2)$. But How (? ) which I am not able to understand. Is there any problem in my work ? Please help.