Dynamical Systems / Differentiatial Equations Problem I need help with my Dynamical systems homework, I honestly don't even really know where to start on it. Any help would be appreciated 
Show that if $x(t)$ is a solution to $x'(t) = x(t)^2 - 1 ,$ then so is $z(t) = -x(t)^{-1}$ and $y(t) = -x(-t).$ What type of symmetries are these and how are they reflected in the phase plane?

As a Riccati equation, you can directly solve this via $x(t)=-\frac{u'(t)}{u(t)}$ where
$$
u''(t)-u(t)=0\implies u(t)=Ae^{-t}+Be^t
\implies 
x(t)=\frac{Ae^{-t}-Be^{t}}{Ae^{-t}+Be^{t}}
$$
Then 
$$
z(t)=-y(-t)=\frac{Be^{-t}-Ae^{t}}{Be^{-t}+Ae^{t}},
$$
has the same form, and so does
$$
y(t)=x(t)^{-1}=\frac{Ae^{-t}-(-B)e^{t}}{Ae^{-t}+(-B)e^{t}}.
$$
Both symmetries combined give a fourth solution
$$
w(t)=y(t)^{-1}=-x(-t)^{-1}=\frac{Be^{-t}-(-A)e^{t}}{Be^{-t}+(-A)e^{t}}.
$$
