Sketch the dynamical system
\begin{align} \dot x_1 = x_2 \\ \dot x_2 = 1 \end{align}
Firstly we may integrate this to find
$$x_2(t) = t + A$$ $$x_1(t) = \frac12t^2+At + B$$
How do I then change this into an equation that I can plot in the $(x_1,x_2)$ plane?
$$ t=x_2-A$$ $$\Rightarrow x_1=\frac{(x_2-A)^2}{2}+A(x_2-A)+B$$ $$\Rightarrow 2x_1=x_2^2 +A^2-2Ax_2+2Ax_2 -2A^2 +2B$$ $$\Rightarrow 2x_1=x_2^2-A^2+2B$$
It is a parabola in $(x_1, x_2)$ plane
$x_1= 1/2(t+A)^2+C= {x_2^2\over 2}+C$, or $x_1-{x_2^2\over 2}=C$, a one parameter family of parallel parabols with horizontal axis.
