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        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
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        34 views

        Why is Jacobi Identity equivalent to holonomy of system? [on hold]

        Or equivalently, why is jacobi identity equivalent to integrability of system? How do I understand it intuitively? Thanks.
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        0answers
        149 views
        +100

        Structural Stability on Compact $2$-Manifolds with Boundary

        I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $2$-manifolds with boundary. Let $M^2$ be a compact connected 2-manifold and $\...
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        37 views

        Solving a system of ODE

        Solve $$\eta_k\frac{d^2C_k}{dz}(z)=-e_k, k = 1,2,3$$ $$C_1(0)=0, C_2(0)=A, C_3(0)=0$$ $$C_1(L)=B, \frac{dC_2}{dz}(L)=0, \frac{dC_3}{dz}(L)=0$$ where $A,B,\eta_k$ some known constant. $e_k, k=1,2,3$ ...
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        0answers
        46 views

        Runge-Kutta 4th order for predator-prey model [on hold]

        I'm trying to compute the numerical solution for a Predator-prey model with 3 equations. This is the model: $$\frac{dx}{dt} =x(1-\frac{x}{k_1})-\frac{pxz}{1+ax+chy}\\ \frac{dy}{dt} =y(1-\frac{y}{k_2})...
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        1answer
        78 views

        Numerical methods for IDE [closed]

        I would like to read a popular literature on the topic "Numerical methods for integro-differential equations". Could you recommend me any articles or book with a brief overview of some methods (maybe ...
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        0answers
        46 views

        Request for help with a tricky Riccati differential equation [closed]

        Long story short, I'm working through a model derivation right now and have arrived at a tricky Riccati Differential Equation that I am afraid has pushed me beyond the limits of my talent. The ...
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        1answer
        96 views

        Trajectory leaving a set

        Consider the differential equation $\dot{x}=f(x)$, where $f: \mathbb{R}^2 \to \mathbb{R}^2$ is smooth. Given a set $A \subset \mathbb{R}^2$, are there some results saying that whenever $x(0) \in A$, ...
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        1answer
        185 views

        Dependence of a solution of a linear ODE on parameter

        Is the following theorem known, or can be easily derived from known results? Consider the differential equation $$w''-kz^{-1}w'=(\lambda+\phi(z))w,$$ where $k>0$ is fixed, $\lambda$ is a large (...
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        41 views

        Solving a non-linear differential equation, generalized Feynman-Kac formula

        I want to solve the following differential equation $f_t(t,\Gamma)+ A(t)f(t,\Gamma) + B(t)f_\Gamma(t,\Gamma)+ \frac{1}{2}tr(\kappa^\intercal \kappa f_{\Gamma\Gamma}(t,\Gamma))+C(t)f_\Gamma (t,\Gamma)\...
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        vote
        1answer
        76 views

        A special oscillatory orbit in space

        Edit: According to the comment of Prof. Eremenko I revise the question. 19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...
        1
        vote
        1answer
        191 views

        Analytical Solution of Two Simultaneous Partial Differential Equations

        I am looking for an analytic solution for the following two equations in the variables $v(x,t)$ and $u(x,t)$: $$ \begin{cases} \dfrac{\partial v}{\partial x} = -m\dfrac{\partial u}{\partial t} \\ \...
        3
        votes
        1answer
        119 views

        Reference request - existence of formal solutions for integrable connections

        Let $K$ be a field of characteristic $0$, let $A = K[[t_1, \ldots, t_n]]$ be a power series ring over $K$, and let $V$ be a free $A$-module. Let $\nabla \colon V \rightarrow V \otimes_A \Omega^1_{A/K} ...
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        0answers
        75 views

        Global solution of second order ODE defined on riemannian manifold

        Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the ...
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        0answers
        136 views

        Analytical solution of a system of nonlinear PDEs

        I am looking for an analytic solution for the equations $$\left\{ \begin{eqnarray} \frac {\partial v} {\partial x} &=& -m \frac {\partial u} {\partial t} \\ \frac {\partial u} {\partial x} &...
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        0answers
        19 views

        Transforming an ODE into a hypergeometric ODE

        I have seen numerous times the statement that any 2nd order ODE with three regular singular points can be converted into a hypergeometric ODE by a change of variables. See for instance https://en....

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