# Sketching a dynamical system

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?

• You can plot the vector field and the integral curves before/without finding the solutions – Marco Disce Mar 26 '16 at 17:05
• I am a little late here but if you are to sketch the dynamical system you should probably heed the comment by @MarcoDisce. If this is an assignment, your teacher may be asking for the vector field and not a particular solution. – John Douma Mar 26 '16 at 17:19

$$t=x_2-A$$ $$\Rightarrow x_1=\frac{(x_2-A)^2}{2}+A(x_2-A)+B$$ $$\Rightarrow 2x_1=x_2^2 +A^2-2Ax_2+2Ax_2 -2A^2 +2B$$ $$\Rightarrow 2x_1=x_2^2-A^2+2B$$
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.