# 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.