Differential equation resembling to cycloid

[tom-73]

What is the function corresponding to this ODE:
http://home.arcor.de/luag/math/dgl.jpg

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

 

[haruspex]

What is the function corresponding to this ODE:
http://home.arcor.de/luag/math/dgl.jpg

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.

 

[tom-73]

Thank you for your comment. I tried to divide 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

 

[haruspex]

Sorry, I overlooked what happens to the RHS. My original suggestion was nonsense.

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.

Ah yes, I see it now. Sorry for the noise.

 

[blue_raver22]

The differential equation shown in the provided image is a second-order ordinary differential equation (ODE) that can be written in complex notation as a*z''(t) + b*|z'(t)|*z'(t) + c = 0, where a, b, and c are constants. This ODE describes the motion of a particle along a cycloid curve, which is a curve traced by a point on the circumference of a circle as the circle rolls along a straight line. The function corresponding to this ODE would depend on the initial conditions of the particle's motion, but it would generally involve trigonometric functions and possibly exponential functions. The numerical solution shown in the graph likely represents the position of the particle over time, with the x-coordinate being the real part of z(t) and the y-coordinate being the imaginary part of z(t). Further analysis and calculations would be needed to determine the specific function that corresponds to this ODE.