A cycloidal pendulum - one suspended from the cusp of an inverted cycloid - is isochronous, meaning its period is constant regardless of the amplitude of the swing. Please find the proof using energy methods: Lagrange's equations (in the images attached to the reply).

Background:
The standard pendulum period of \(2\pi\sqrt{L/g}\) or frequency \(\sqrt{g/L}\) holds only for small oscillations. The frequency becomes smaller as the amplitude grows. If you want to build a pendulum whose frequency is independent of the amplitude, you should hang it from the cusp of a cycloid of a certain size, as shown in the gif. As the string wraps partially around the cycloid, the effect decreases the length of the string in the air, increasing the frequency back up to a constant value.

In more detail:
A cycloid is the path taken by a point on the rim of a rolling wheel. The upside-down cycloid in the gif can be parameterized by \((x, y)=R(\theta-\sin\theta, -1+\cos\theta)\), where \(\theta=0\) corresponds to the cusp. Consider a pendulum of length \(L=4R\) hanging from the cusp, and let \(\alpha\) be the angle the string makes with the vertical, as shown (in the proof).

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L'Hôpital's rule is when you're trying to calculate the derivative of a complex function. Not only you fail, but the function beats you up, steals your lunch money, and sends you to the hospital.

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Toolbox Website
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Important Dates
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WHY VARIATION OF PARAMETERS WORKS

This is more conceptual than a proof, but I find it comforting.

Consider the equation:

y' - y = xe^x

The solution will consist of a Homogeneous Solution and a Particular Solution; add the two for the complete solution.

The Homogeneous Solution is the solution that, when you run it through the left side of the equation, it always goes to zero. So you need the Homogeneous Solution for the same reason you need the "+ C" in antiderivatives: even if it's kind of the boring throwaway part of the solution, it is still part of the complete solution.

But Variation of Parameters finds another use for the Homogeneous Solution. Let us suppose that the Particular Solution is the Homogeneous Solution times a function "u". Well, if you feed the Particular Solution through the left side of the equation, the Homogeneous Solution part will tend to go away, leaving "u". So then you can multiply "u" by the Homogeneous Solution, and you've got your Particular Solution.

It kind of reminds me of how Taylor Series work. In a Taylor Series, you have a function that you can think of as secretly containing a multitude of polynomial terms, and the trick is finding a way to torture the function into confessing the coefficients on each polynomial term. In the case of Taylor it's done by iteratively differentiating and then setting "x" to zero, thus leaving a constant that is the coefficient for a given polynomial. (There's also that "n!" term but that's just details.)

Or Fourier Series: a periodic function secretly contains a multitude of sine and cosine terms, and again you find a way to torture it into confessing the coefficients on each sine / cosine. In that case the torture technique involves integration.

And in the case of Variation of Parameters, the torture technique is, we know what part of the particular solution gets burned away by the left side of the equation; the charred skeleton that remains is the other part of the particular solution.

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Convection–diffusion equation
The convection-diffusion equation is a more general version of the scalar transport equation. It is a combination of the diffusion and convection (advection) equations. It describes physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection.
\[\dfrac{\partial c}{\partial t} = \mathbf{\nabla} \cdot (D \mathbf{\nabla} c - \mathbf{v} c) + R\]

\[\dfrac{\partial c}{\partial t} = \underbrace{\mathbf{\nabla} \cdot (D \mathbf{\nabla} c)}_{\text{diffusion}}-\overbrace{\underbrace{\mathbf{\nabla}\cdot (\mathbf{v} c)}_{\text{advection}}}^\text{convection} + \overbrace{\underbrace{R}_\text{destruction}}^\text{creation}\]

\(\mathbf{\nabla} \cdot (D \mathbf{\nabla} c)\) is the contribution of diffusion.
\(- \mathbf{\nabla}\cdot (\mathbf{v} c)\) is the contribution of convection or advection.
\(R\) describes the creation or destruction of the quantity.

where
\(c\) is the variable of interest.
\(D\) is the diffusivity.
\(\mathbf{v}\) is the velocity field, and
\(R\) is the sources or sinks of the quantity \(c\).

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New lectures on undergraduate differential equations:

18. Mechanical vibrations, part 2: free damped motion (Notes on Diffy Qs, 2.4)
https://youtu.be/Z6BR9aDPdy4

19. Nonhomogeneous equations, part 1: undetermined coefficients (Notes on Diffy Qs, 2.5)
https://youtu.be/jRPuCCQqGXU

Based on the free book: Notes on Diffy Qs https://www.jirka.org/diffyqs

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