how to find zero solutions and stability? Show that the zero solution of $$ \large \ddot{x}+bx^2 \dot{x}+kx=0$$
is asymptotically stable if $b>0$ and unstable if $b<0$. Does this depend on the sign of $k$?
I know for 1st order equation but how to find stability of 2nd order equation?
Can I convert it into system of 1st order equations as follows:
Let $y=x$ and $z=\dot x$.  Then,
$\dot z=\ddot x=-bx^2 \dot x-kx=-by^2z-ky,$ 
i.e., $\dot z=-by^2z-ky, ......(1)$
and $ \dot y=z, .......(2)$
These are the two equations.
But how to find zero solutions and stability ?
Do we need to linearize this system again?
Help me

Any constant solution obviously has $\dot x=0$, $\ddot x=0$ so that the equation $kx=0$ remains.
For $k<0$ you get a saddle point, thus $k>0$.
For the stability consider
$$
\frac{d}{dt}\frac12(\dot x^2+x^2)=-bx^2\dot x^2
$$
which tells you in which direction the solutions cross the circles around the origin.
