Differential equation resembling to cycloid

In summary, the conversation discusses the function corresponding to a given ODE in complex notation, which is derived by multiplying the second equation by i and adding it to the first equation. The ODE describes a cycloid and a phugoid, but the phugoid does not have an analytic solution.
  • #1
tom-73
2
0
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
 
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  • #2
tom-73 said:
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.
 
  • #3
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
 
  • #4
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.
 
  • #5


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.
 

1. What is a differential equation resembling to cycloid?

A differential equation resembling to cycloid is a type of mathematical equation that describes the shape of a cycloid, which is a curve traced by a point on the circumference of a circle as it rolls along a straight line. This type of differential equation involves derivatives of the cycloid curve and is often used in physics and engineering to model various motion and oscillation phenomena.

2. What is the significance of the cycloid curve in this type of differential equation?

The cycloid curve is significant because it is a solution to the differential equation resembling to cycloid. This means that the curve satisfies the equation and can be used to describe the behavior of various systems in motion, such as a pendulum or a rolling wheel.

3. How is a differential equation resembling to cycloid different from other types of differential equations?

A differential equation resembling to cycloid is different from other types of differential equations because it involves a cycloid curve as a solution, rather than a traditional mathematical function. This makes it more complex and challenging to solve, but it also allows for more accurate modeling of real-world phenomena.

4. What are some applications of differential equations resembling to cycloid?

Differential equations resembling to cycloid have many applications in physics and engineering. They can be used to model the behavior of simple pendulums, rolling wheels, and other systems in motion. They are also used in fields such as fluid dynamics, electrical circuits, and economics to describe various phenomena and make predictions.

5. Is there a specific method for solving differential equations resembling to cycloid?

There is no one specific method for solving differential equations resembling to cycloid, as the approach may vary depending on the specific equation and its applications. However, some common techniques used for solving differential equations, such as separation of variables, can also be applied to this type of equation. Other methods, such as using numerical approximations, may also be utilized.

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