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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.

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        If f(x) and g(x) are two functions such that f'(x) = g(x) and g'(x) = f(x), prove that f''(x) - g''(x) is a constant [on hold]

        If f(x) and g(x) are two functions such that f'(x) = g(x) and g'(x) = f(x) for all x, prove that f''(x) - g''(x) is a constant
        10
        votes
        0answers
        200 views

        Computing spectra without solving eigenvalue problems

        There is a rather remarkable conjecture formulated in this paper, "Computing spectra without solving eigenvalue problems," https://arxiv.org/pdf/1711.04888.pdf and in this talk by Svitlana Mayboroda ...
        0
        votes
        0answers
        26 views

        Question on the existence of Carathéodory solutions to a (scalar) first order discontinuous ODE

        Consider the scalar i.v.p. in ${\mathbb R}$ $$ x'=f(t,x), \; t\in[0,T], \; x(0)=x_0 $$ where $T\in {\mathbb R}$, $T>0$, $x_0\in {\mathbb R}$, and $f:[0,T] \times {\mathbb R}\mapsto {\mathbb R}$ ...
        4
        votes
        1answer
        86 views

        A non-geodesible foliation of $S^3$ or $S^2\times S^1$

        Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation? If the answer is ...
        1
        vote
        0answers
        122 views

        Lax Milgram for non coercive problem?

        I obtained the variational form of my problem. and the bilinear form is below. Bilinear Form Let $\Omega\subset\mathbb{R}$ be an open set. For $u,v\in H^1_0(\Omega)$. I have $$a(u,v)=\int_\Omega u(t)...
        1
        vote
        0answers
        62 views

        Geometric interpretation of Lagrange's linear equation and its solution [closed]

        What is the geometric meaning of the Lagrange's linear equation, $$P\dfrac{\partial z}{\partial x} + Q \dfrac {\partial z}{\partial y}= R \\Pp+ Qq= R$$ where $P$,$Q$,and $R$ are functions of $x\,$,$\,...
        0
        votes
        1answer
        53 views

        Backward Stochastic Differential Equation

        Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and $$ dX_t=f_tdt+B_tdW_t $$ where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...
        2
        votes
        0answers
        13 views

        Is there an extension of the Kovacic algorithm to handle algebraic coefficients?

        Kovacic's algorithm solves second-order linear homogeneous differential equations with rational function coefficients. I'm wondering if anybody has extended this algorithm to handle algebraic ...
        1
        vote
        1answer
        139 views

        Nonlinear second order ODE $y''+f(x)y=g(x)y^3$

        I encountered the following ODE in order to find a solution for Einstein equation $$y''+f(x)y=g(x)y^3.$$ It seems to me that it is not among the solvable nonlinear second order differential equations....
        1
        vote
        0answers
        78 views

        Solution of $s\zeta'(s)-\zeta(s)=0$ for some infinite strip $\subset\{0<\Re s<1\}$

        I did the following calculation, first I take the $k$-th derivative with respect $s$ of the Mellin transform of the fractional part $\{\frac{1}{t}\}$ defined for $0<\Re s<1$ as it is showed in ...
        1
        vote
        0answers
        27 views

        How to solve or analyse the smallest eigenvalue of 2 coupled 1st-order linear ODEs?

        I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities \begin{align} -\mathrm{i} u'(x) +f^*(x) ...
        2
        votes
        2answers
        243 views

        Asymptotics of solutions to the ODE $f''(t)-e^{-2t} f(t)=0$

        Consider the ode $$ f''(t)-e^{-2t} f(t)=0. $$ What is the general behaviour of $|f|$ for large $t$s? Is it true that there exists an $A\in \mathbb{R}$ and there exists a positive $B$ such that $|f(...
        0
        votes
        1answer
        147 views

        Solution of nonlinear second-order ODE $y''+\frac{(y'+2ax)^2+4b^2}{2y}+\frac{10}{3}a=0$

        Is there any way of solving the following second-order ODE $$y''+\frac{(y'+2ax)^2+4b^2}{2y}+\frac{10}{3}a=0,$$ where $a$ and $b$ are some constant? If we know that one solution exists, how would it ...
        2
        votes
        0answers
        47 views

        Second order non-instantaneous impulsive evolution equations

        The first order linear non-instantaneous impulsive evolution equations is given as; $u'(t)=Au(t)~~ t\in[s_i,t_{i+1}],\,i\in\mathbb{N_0}:=\{0,1,2,...\}$ $u(t_i^+)=(E+B_i)u(t_i^-),\,\,i\in\mathbb{N}:=...
        0
        votes
        0answers
        38 views

        Coexistence of different solutions in a nonlinear matrix differential equation

        I've faced a system of first-order nonlinear matrix differential equation, and I have tried to use perturbation method to approach the solutions. The differential equation has the form: \begin{...

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