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Solving the linear first-order $$ h'=h K^{1/2}e^{f/2}/\sqrt 2 \tag{$\ast$}$$ makes the rhs to vanish. We get then \begin{align} 2h''-f'h'-Ke^f h&=h' K^{1/2}e^{f/2}\sqrt 2+h K^{1/2}e^{f/2}f'(\sqrt 2)^{ …

I understand that your ODE is one-dimensional: in that case you can avoid Lipschitz continuity and replace it by transversality: The autonomous equation $$ \dot x =f(x),\quad x(0)=x_0, $$ has a uniqu …

Forget about finding a closed analytical expression for the fundamental matrix. Think about the simple case of a second order scalar equation such as the Airy equation $\ddot x=tx.$ There is neverthe …

A simple local existence theorem in a Banach space setting. Let $E$ be a Banach space, $I$ be an interval of $\mathbb R$, $\Omega$ be an open subset of $E$, $F:I\times \Omega\longrightarrow E$ be a co …

Overkill...If $x,y$ are solutions, we get $$ \vert x(t)-y(t)\vert\le\vert x(0)-y(0)\vert+ \int_{0}^{t}\Vert A(s)\Vert\vert x(s)-y(s)\vert ds=R(t) $$ and $ \dot R(t)=\Vert A(t)\Vert\vert x(t)-y(t)\vert …

Let me work with $n$ dimensions: you want to study the vector field $$ X=\sum_{1\le j\le n} a_j(x)\frac{\partial}{\partial x_j}, \tag {1}$$ and in particular find the so-called first integrals of $X$ …

Let $Ai$ be the classical Airy function and let $(a_j)_{j\ge 1}$ be the strictly decreasing sequence of its zeroes: we have $a_{j+1}<a_j<\dots <a_2<a_1<0$, $\lim_{j\rightarrow +\infty}a_j=-\infty$. I …

The operator $\frac{\partial}{\partial t}+t\Delta_x$ is not locally solvable near $t=0$. It could be seen as a quasi-homogeneous version of Mizohata operator $D_t+it\vert D_x\vert$ since its symbol is …

Your equation falls in the category of regular singular differential equation. Writing your equation as $$ x z'=a z+g(x,z), \tag{$\ast$}$$ the singularity is called regular because the exponent of th …

I want to consider the solutions of the following fourth-order ODE: $$ f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0, \tag{$\ast$}$$ where $a,b$ are complex parameters. It turns out that with a Fourier transforma …

Let $P$ be a pseudodifferential operator with symbol $p(x,\xi)$ belonging to $S^m_{1,0}$ and let $a(x)$ be a $C^\infty$ function. Then the operator $Pa$ defined by $Pa u=P(au)$ is a pseudodifferential …

Let us work in $\mathbb R^n$. The distribution solutions of the equation $$ (x\cdot \partial _x) u=0 $$ are the distributions which are homogeneous of degree 0. Here $x\cdot \partial_x=\sum_{1\le j\le …

I would like to give another example, a very classical one. Consider a smooth open subset $\Omega$ of $\mathbb R^n$ and let $T$ be a smooth vector field tangential to the boundary $\partial\Omega$. Th …

(1) Lagrangean description. Let us consider a $N\times N$ system of autonomous ODE: $$ \dot x=a(x),\quad \mathbb R\ni t\mapsto x(t)\in \mathbb R^N,\quad a:\mathbb R^N\rightarrow \mathbb R^N. $$ Instea …

Consider the (stationary) Schrödinger equation $$ -\Delta u+ Vu=0, $$ or the differential inequality $\vert\Delta u \vert\le \vert V u\vert$, where $V$ is some "potential" function. The following uniq …