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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

2 votes
Accepted

(Non-) Convergence of solutions in a family of linear ODE's

Using the expansion for $\arctan(t)$ as $t\to +\infty$ in the form $$\arctan(t) = \frac{\pi}{2}-\frac{1}{t}+\frac{1}{3t^3}+\cdots$$ we get, for fixed $x>0$ and positive $d\to 0$: $$ f_d(x) \sim \frac{ …
Loïc Teyssier's user avatar
0 votes

Closed field lines in the plane

The answer to both question is «Yes». The OP is asking what kind of singularity the foliation of the Riemann sphere induced by the differential equation in Alexandre's answer possesses. It can be rewr …
Loïc Teyssier's user avatar
6 votes

Non-linear first order ODE

Setting $x:=z^{-1}$ and $f:=e^u$ your differential equations is transformed into $$z^2\mathrm{d}u=(zu-e^{-u})\mathrm{d}z$$ The curves integrating the above differential relation define a regular fol …
Loïc Teyssier's user avatar
19 votes
Accepted

Solution of linear ODE

This is known as Differential Galois Theory, first developed by Picard and Vessiot. In your case you should look for authors such as Kolchin or Singer and Van Der Put. Some systems definitely admit s …
Loïc Teyssier's user avatar
5 votes

Abstract ODE; PDE; uniqueness of solution

It is true if $A$ is assumed bounded (as can be found in Section 2 of the reference given by Andras Batkai). An "elementary proof" consists in proving the fundamental theorem of calculus, that is for …
Loïc Teyssier's user avatar
3 votes

Difference equation and formal series

In general, this question makes sense at a formal level near $\infty$ only, i.e. for power series involving negative powers of $x$. Setting $z:=\frac{1}{x}$, the equation becomes a so-called homologic …
Loïc Teyssier's user avatar
3 votes
Accepted

Exact Differential Equations of Order n via Pfaffian Differential Equations?

I'll assume all along you're referring to ODE with real-analytic/holomorphic coefficients. You're looking for something called "non-linear differential Galois theory". This is related to this question …
Loïc Teyssier's user avatar
8 votes
Accepted

Examples of Stokes data

It helps to understand irregular singularities as the merging of regular singular points, say $$(x^2-a^2)y'+y=0$$ as $a\to0$. For nonzero $a$ the data is encoded as monodromy (constant) matrices actin …
Loïc Teyssier's user avatar
3 votes
Accepted

A theory of bifurcation of braids ?

Here is a couple of references regarding the holomorphic world, although they do not represent an answer to your question per se (but I'm afraid this is too long a comment, and might anyhow interest p …
Loïc Teyssier's user avatar
1 vote

Dependence of a solution of a linear ODE on parameter

Consider the analytic vector field $$X(z,w,W)=z \partial_z+zW\partial_w+(kW+z(\lambda+\phi(z))w)\partial_W$$ whose orbits project to the graphs of solutions $z\mapsto w(z)$ (it is simply the companion …
Loïc Teyssier's user avatar