# Questions tagged [differential-equations]

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

1,041 questions
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### Has this Peculiar Property of Unit Circles Already been Noticed?

Yesterday I needed to do some calculations with circles and "ventured" to calculate the arc length via the $\int{\sqrt{1+\left(f'(x)\right)^2}}$ formula and was baffled to see that in the case of unit ...
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### Analytic solutions to algebraic differential equation

Dear Colleagues and Friends, Here I need to find some good reference on a subject that seems very much studied: sorry, if the rest of this question is too naive. I believe that it's known that if a ...
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### Existence of a solution to a boundary value problem

Consider an ODE of the form $$y''=f(x,y,y')$$ with $x \geq 0$, and initial conditions of the form $$y(0)=y_0>0, \\y'(0)=m_0.$$ I want to claim that under reasonable conditions (which I can't ...
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### Regular holonomic D-modules as generalisation of regular singular points

I'm trying to understand why the definition of a regular holonomic D-module is a good generalisation of the usual definition of a regular singular point for a differential equation. More precisely, ...
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### Wolff's article: Note on counterexamples in strong unique continuation problems

I am reading Wolff's Note on counterexamples in strong unique continuation problems: http://www.ams.org/journals/proc/1992-114-02/S0002-9939-1992-1014648-2/S0002-9939-1992-1014648-2.pdf On Page 3, ...
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### Continuation (Extension) of Harmonic Functions

Suppose $(M,g)$ is a simply connected smooth Riemannian manifold with smooth boundary and suppose that $U \subset \partial M$ is a smooth open connected subset of the boundary. Now my question is the ...
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### How can I show the principal symbol is not elliptic?

Let us assume the principal symbol of a nonlinear differential operator $E$ at a point $p$ is $$\sigma‎_{‎E‎}(p):‎\Gamma (‎T^*M‎)\times \Gamma (T^*‎M‎)\to \Gamma (T^*‎M‎)‎$$‎ which acts as follows: ‎‎‎...
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Consider $V(x)$ a one dimensional polynomial, confining, symmetric double well potential i.e. $V(x)=V(-x)$ for all $x\in\mathbb{R}$ $\displaystyle \lim_{x\to\pm\infty} V(x)=+\infty$ V(x)\in \mathbb{... 0answers 199 views ### Solutions for an ODE I would like to know if someone could provide me the solution(s) for the following equation: \begin{align*} \frac{d^{2}w}{dx^{2}}\cdot\frac{dw}{dx} = e^{-x}w \end{align*} Wherew > 0$,$w''>0... 0answers 62 views ### Well-posedness of a differential equation: Cauchy vs Dirichlet [closed] I am struggling to understand what makes some differential equations well-posed Cauchy (initial value) problem rather than a Dirichlet (boundary value) problem. Consider, for example, a simple ... 1answer 164 views ### Riemann surface from Riccati equation I have quite a practical question motivated by physics. Consider the Riccati equation whose solution gives a quantum-mechanical (QM) analogue of the classical momentum: $$(p(x))^2 + \dfrac{\hbar}{i}... 2answers 275 views ### On Wilson's claim that Lyapunov function level sets are not exotic spheres In Wilson's paper "The structure of the level surfaces of a Lyapunov function," he states in Corollary 1.3 that the level sets of a smooth Lyapunov function are diffeomorphic to a standard sphere. (... 0answers 494 views ### What is a Green's function in the language of \mathcal{D}-modules? Let P be an analytic linear differential operator defined on some open interval X=(a,b) and \mathcal{M}=\mathcal{D}_X / \mathcal{D}_X \bullet P the corresponding \mathcal{D}-module. I'm trying ... 3answers 640 views ### Vector field with holomorphic flow Let (M,J) be a complex manifold. Suppose that X is a real vector field such that the flow of X is by biholomorphisms.Question Show the flow of JX is by biholomorphisms. I know one reference ... 1answer 342 views ### Fredholm index vs. Limit cycle theory Let A be the algebra of all smooth functions f: \mathbb{R}^2 \to \mathbb{R} such that f is flat at the origin and is real analytic on \mathbb{R}^2 \setminus \{0\}. Let B be ... 2answers 445 views ### Is there a closed-form solution for \frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}? I am looking for an exact solution for the following special case of Chini Equation with 2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+,$$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$I ... 1answer 160 views ### Reference request: Original source of Yosida approximation Numerous papers/books(citation needed) refer to the operator$$A_\lambda := \lambda AR_\lambda (A) = \lambda^2 R_\lambda(A) - \lambda I$$where R_\lambda(A)=(I+\lambda A)^{-1} is the resolvent, as ... 2answers 509 views ### An ordinary differential equation While I was working on a variational problem, I met this equation as its Euler-Lagrange equation, but I cannot solve it: x= \frac{af'(x)}{\sqrt{1+af'(x)^{2}}} + \frac{bf'(x)}{\sqrt{1+bf'(x)^{2}}} \ (... 1answer 261 views ### Gauge equivalence between operators I have tried to figure out the following problem for some time now, but with little success: Let \mathcal{L} be a third order linear differential operator with coefficients in \mathbb{C}(X) . ... 0answers 215 views ### Are all solutions to an ordinary differential equation continuous solutions to the associated implied differential equation and vice versa? Now I have to heavily emphasize the fact that I have never studied differential algebra or the concept of other types of differentiation (which is what I believe is the concept behind a differential ... 0answers 35 views ### A uniqueness result for a BVP over a semi-infinite interval Let q:[0,\infty) \to \mathbb{R}, and consider the ODE$$u''(x)=q(x)u(x) $$with the boundary conditions u(0)=\lim_{x \to \infty} u(x)=0. Under what conditions on q is u \equiv 0 the only ... 0answers 285 views ### On modified Bessel solutions to complex ODE's using Kummer's series I am trying to reduce the following ODE to Bessel's ODE form and hence solve it:$$x^{2}y''(x)+x(4x^{3}-3)y'(x)+(4x^{8}-5x^{2}+3)y(x)=0\tag{1} \, .I tried to solve it via the standard method, i.e.,... 0answers 34 views ### Uniqueness of positive solutions to the n-vortex type equation The n-vortex equation, in the context of optical vortex solitons, is of the form -(ru_{r})_r+\dfrac{n^2}{r}u+\beta ru= f(u^2)ru,\quad r\in (0,\infty),\\ u(0)=0=u(\infty), \end{... 1answer 91 views ### Does this equation have an explicit solution? [closed] For a positive constant C: \begin{align} y(x)+C\ln y(x)=f(x). \end{align} At least from specific f(x), such as piece-wise linear function, is there an explicit solution for y(x)? 2answers 206 views ### Minimum eigenvalue of One-dimensional Schrodinger Operator Consider the One dimensional Schrodinger Operator -\frac{d^2}{dx^2} + V(x) $$Where the Potential Function V is of the form V(x) = x^4 - a^2x^2 , a \in \mathbb{R} . Now of course,the ... 0answers 64 views ### Convergence of Bessel (Sturm-Liouville) Expansions at the End Points I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ... 1answer 136 views ### A non vanishing vector field on S^3 with a periodic attractor Is there a non vanishing real analytic vector field X on S^3 such that X has an attractor periodic orbit(An asymptotically stable periodic orbit) ? What about the smooth case? 0answers 189 views ### Does the divergent solution of this equation :f'=e^{f^{-1}} of Gevrey type and could be Borel summation applied for it? This question was asked here in MO by someone seeking for the solution of the functional -differential:f'=e^{f^{-1}} not exactly an O.D.E, and again here seeking for the growth rate of it solution ... 1answer 150 views ### Solutions to linear SDE with many noise sources It is well known how to solve the linear stochastic ODEs with one source of noise$$dX_t=(a(t)X_t+c(t))dt+(b(t)X_t+d(t))dW_t$$See, for instance, https://math.stackexchange.com/questions/1788853/... 1answer 216 views ### Hörmander's hypoellipticity theorem for complex coefficients Hörmander's theorem says that if L = \sum _{i=1} ^r X_i ^2+ X_0 + f on some open subset U \subseteq \Bbb R has the property that the Lie algebra generated by \{X_0, \dots, X_r\} at every point ... 0answers 87 views ### A global geometric formulation of the fundamental theorem of Picard Vessiot theory? Let X be a smooth curve over an algebraically closed field of characteristic 0. The category of quasi-coherent D-modules \mathcal{D}_X-Mod is a symmetric monoidal abelian category. We can ... 0answers 117 views ### Geometric interpretation of energy-momentum tensor and Lagrangian associated to a soliton equation I have a question for you. BACKGROUND Consider an immersion F\,:\;\mathscr S\;\longrightarrow\;\mathbb E^3 of a surface \mathscr S in the 3-D euclidean plane \mathbb E^3 with canonical ... 1answer 198 views ### Spectral growth of One dimensional Schrodinger Operator Conside the One dimentional Schrodinger Operator$$ -\frac{d^2}{dx^2} + ( V(x) + E ) $$Where the Potential Function V is of the form V(x) = ax^2 + b^2x^4 , a,b \in \mathbb{R} . What is known ... 0answers 147 views ### Is the closed orbit of the Vander pol equation a stable periodic orbit? We consider the Vander Pol vector field$$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$on$\mathbb{R}^2.$It is well known that this equation has a unique limit ... 0answers 49 views ### Control of the Hessian of the distance function at infinity Let$(M,g)$be a complete noncompact manifold and$r$is the distance function to a fixed point. We assume that the sectional curvature$|Rm|(x) \le K(r(x))$where$K(r)=\frac{1}{1+r^k}$for some$k&...
I am trying to evaluate an indefinite integral of the form $\int \frac{dz}{A u_1^2 + Bu_2^2 + Cu_1u_2}$ where $u_1$ and $u_2$ are two independent solutions to the ODE $u'' + F(z)u = 0$ This ...