# Questions tagged [differential-equations]

Ordinary or partial differential equations. Delay differential equations, neutral equations, integro-differential equations. Well-posedness, asymptotic behavior, and related questions.

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### Zero-curvature formulation of the Camassa-Holm hierarchy

In the book of Gesztesy and Holden (see the following article of the same authors), they state that the (stationary) Camassa-Holm hierarchy may be cast as a zero-curvature equation \begin{align} -V_{n,...
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### An ODE is linear if and only if the maximal solutions are a linear space?

Let $I$ be a non trivial interval of $\mathbb R$, let $f : I \times \mathbb R^n \to \mathbb R^n$ and consider the following ordinary differential equation (ODE): \begin{equation}\tag{$\mathscr E$}\...
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### Identity principle of solutions of SL-problems with matching values on open set

Situation (cut short): Corresponding solutions (by eigenvalue) of two given regular Sturm-Liouville problems with homogeneous Neumann BC, same spectrum but possibly distinct coefficient functions, &...
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### $f' = e^{f^{-1}}$, a third time

I am of the impression the differential equation $f' = e^{f^{-1}}$ was considered on mathoverflow for the first time here: How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$? It was found ...
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### Eigenvalues of Sturm–Liouville operator

Can we calculate the eigenvalues and eigenfunctions of the following operator in $W^{1,2}(\mathbb{R})$? $$-\left(\frac{1}{\cosh^2x}\right)y''-\frac{2}{\cosh^4x}y=\lambda y.$$
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### Is the function $F(x) = \exp(x) + \exp(\exp(x))x$ a hypertranscendental function?

The function $F(x) = \exp(x) + \exp(\exp(x))x$ plays a role in the formulation of the Lagarias inequality: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$ If we put $x = \log(H_n)$, then this inequality ...
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### Show that an integral operator with Bessel function kernel is bounded on $L^2(0,\infty)$

Let $J_0$ denote the Bessel function of the first kind of order $\nu = 0$ (see DLMF 10.2), $$J_0(z) = \sum_{k = 0}^\infty (-1)^k \frac{(\tfrac{1}{4} z^2)^k }{k! \Gamma(k + 1)}.$$ Put \$u_0(r) = r^{1/...
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### Factoring higher-order differential operators

I have been researching various methods for solving differential equations. In particular, I want to better understand the factoring approach. For example, if we want to solve a general second order ...
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### Harmonic function over a square with linear Neumann boundary conditions

For a rectangle with height 1 and length 2, here is the unique numerical solution (showing contours of the equipotential from 0, defined by the bottom, to 0.54, the numerically-calculated maximum) to ...
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