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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
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 ...
0answers
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### 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 $f:[0,T] \times {\mathbb R}\mapsto {\mathbb R}$ ...
1answer
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### 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 ...
0answers
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### 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}:=...
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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