Question: phase diagram

Hi
How to get from the differential equation:
diff(x(t), t, t)+c*(diff(x(t), t))+k*sin(x(t)) = 0;
print(`output redirected...`);               / d  / d      \\     / d      \                 
              |--- |--- x(t)|| + c |--- x(t)| + k sin(x(t)) = 0
              \ dt \ dt     //     \ dt     /                 
which transforms a differential equation of the phase trajectories
DEtools[convertsys](diff(diff(x(t), t), t)+c*(diff(x(t), t))+k*sin(x(t)) = 0, [], [x(t)], t, x, y);
[                                             [                     d      ] 
[[y[1] = x[2], y[2] = -c x[2] - k sin(x[1])], [x[1] = x(t), x[2] = --- x(t)],
[                                             [                     dt     ] 
               ]
  undefined, []]
               ]
subs([[y[1] = x[2], y[2] = -c*x[2]-k*sin(x[1])], [x[1] = x(t), x[2] = diff(x(t), t)], undefined, []][1], [Diff(x[1], t) = y[1], Diff(x[2], t) = y[2]]);
             [ d                d                              ]
             [--- x[1] = x[2], --- x[2] = -c x[2] - k sin(x[1])]
             [ dt               dt                             ]
  and with specific points (0, 0), (± PI, 0), (± 2Pi, 0) get to show the following phase diagram

 

 

 

 

 

 

 

HerClau

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