3.34 How to make phase plot of second order ODE?

Make a phase plot of

\[ \frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {\frac {d}{d t}x \left (t \right )}{2}+x \left (t \right ) = u \left (t \right ) \]

By plotting \(x(t)\) vs \(x'(t)\) without solving the ODE.

restart; 
alias(DS=DynamicSystems): 
ode := diff(x(t),t$2) +1/2*diff(x(t),t)+ x(t) = u(t); 
sys:=DS:-DiffEquation(ode,'outputvariable'=[x(t)],'inputvariable'=[u(t)]); 
sys0:=DS:-StateSpace(sys); 
eq1:=diff(x1(t),t)=sys0:-a[1,..].Vector([x1(t),x2(t)]); 
eq2:=diff(x2(t),t)=sys0:-a[2,..].Vector([x1(t),x2(t)]); 
DEtools:-DEplot([eq1,eq2],[x1(t),x2(t)],t=0..35,[[x1(0)=1,x2(0)=1]],x1=-2..2,x2=-2..2, 
        numpoints=200, linecolor=black, axes=boxed);
 

Figure 4: Phase Plot