## Differential equation resembling to cycloid

What is the function corresponding to this ODE:

In complex notation it obviously shows up like this:

a * z''(t) + b * |z'(t)| * z'(t) + c = 0;

The numerical solution shows a graph resembling to a cycloid.

Thanks for any help!
Tom

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 Quote by tom-73 What is the function corresponding to this ODE: In complex notation it obviously shows up like this: a * z''(t) + b * |z'(t)| * z'(t) + c = 0;
It does? I don't see how.
If you divide through by the surd and subtract the 1st eqn from the second, I believe you get something integrable.

 Thank you for your comment. I tried to devide and subtract. The problem is the term in the middle: (y' - eps*x') vs. (x' + eps*y') It makes the situation even worse - I did not succeed in finding a simplified pattern. The complex notation in my first post has been derived by simply multiplying the 2nd equation by i and adding the result to the first equation. What I investigated in the meanwhile: The cycloid ODE in complex notation should be a * z''(t) + b * z'(t) + c = 0; The only difference is the multiplication with |z'| in the middle which in fact produces a value near 1 for curtate cycloids with r1 << r0 (the point tracing out the curve is inside the circle, which rolls on a line AND it is close to the center). The ODEs in my first posts describe a phugoid, a more general form of the cycloid I suppose. It seems that the phugoid has no analytic solution. Any suggestions? Tom

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