# Questions tagged [differential-equations]

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

1,093
questions

**-6**

votes

**0**answers

74 views

### 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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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{...