Writing nonautonomous systems as autonomous systems Apparently any mth order nonautonomous system is equivalent to a first order autonomous system in higher dimensional space.
How does this work in practice? I would you write $\displaystyle \frac{d^3x}{dt^3}=sin(t) \frac{x\ddot{x}}{\dot{x}^2}$ as a first order autonomous system on $\mathbb{R^4}$. I would like hints on how to start this question and no full solutions please.

The usual way is to define $z$ in $\mathbb R^4$ by $$z=(t,x,\dot x,\ddot x)$$ and to note that $x'''=A(t,x,\dot x,\ddot x)$ for some function $A:\mathbb R^4\to\mathbb R$ if and only if $\dot z=B(z)$ where the function $B:\mathbb R^4\to\mathbb R^4$ is defined by $$B(z_1,z_2,z_3,z_4)=(1,z_3,z_4,A(z_1,z_2,z_3,z_4))$$
