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2 votes

Converting an algebraic equation into a ODE

Disclaimer: I am not sure I understand what you mean by "algebraic equation". I don't know if you really mean functions that are expressible as roots of a polynomial equation. If that's the ...
Willie Wong's user avatar
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1 vote

Method of characteristics for higher order PDEs in more than two variables

Check Harry Bateman’s book, Partial Differential Equations, section 2.24, pg. 133. The book is on the internet archive web site. He gives an equation for the characteristics for a second order PDE in ...
wijohns777's user avatar
4 votes
Accepted

Existence and uniqueness of solutions to a distributional ordinary differential equation

This second answer specifically addresses the issue of trying to interpret $v\circ x$ when $x$ is not regular and $v$ is a distribution. I will tie this into the edit to illustrate the problem. The ...
Willie Wong's user avatar
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6 votes

Existence and uniqueness of solutions to a distributional ordinary differential equation

Let me explain several reasons why I think what you are trying to do cannot work. Algebra Suppose, for a moment, that $v(x(t))$ makes sense as a pull-back distribution. Then we would be allowed to ...
Willie Wong's user avatar
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1 vote

Is there any work on distributional vector fields?

The specific question you ask here is basically about the existence of distributional antiderivatives. The answer is "yes", and that you can also recover the "$+C$". The main ...
Willie Wong's user avatar
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1 vote

Is there any work on distributional vector fields?

Apart from signs, aren't you just asking for a function/distribution whose distributional derivative is $\delta$ (or $\delta+1$, or whatever)? Those integrals are by-abuse-of-notation just expressions ...
paul garrett's user avatar
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5 votes
Accepted

The stability of the equilibria of a non-linear ODE system

The first two equations have an integral of motion: $$ c \log(p)-\gamma f-\gamma p=\mathrm{const}. $$ This allows one to exclude $f$, and then there are 2 eqs with 2 variables $p$ and $T$. They can be ...
Martin Nicholson's user avatar
0 votes

Non-integrability of Abel's equation

The paper by Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz and Chara Pantazi Differential Galois theory and non-integrability of plane polynomial vector fields, J. Differential ...
Phil Harmsworth's user avatar
5 votes

Frobenius theorem and the size of integral manifold

Your equations are equivalent to the $1$-form equations $$ \mathrm{d}f = X_0(f,g)\,\mathrm{d}s + Y_0(f,g)\,\mathrm{d}t \quad \text{and}\quad \mathrm{d}g = X_1(f,g)\,\mathrm{d}s + Y_1(f,g)\,\mathrm{...
Robert Bryant's user avatar
3 votes

Generalized Fuchsian-type PDE?

Here's one way to get the hypergeometric function for the "simpler" equation: Consider the operator $x^3 (1 + t\partial_t)(\partial^3_{xxx} + \frac{6}x \partial^2_{xx} + \frac{6}{x^2} \...
Willie Wong's user avatar
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5 votes

Generalized Fuchsian-type PDE?

In your simplified case, I don't see how $A(x,0) = 1$. In fact, the overall factor of $t$ should for the solution to vanish for all $x$ at $t=0$. Actually, I think there is no solution to your ...
Igor Khavkine's user avatar
3 votes

Generalized Fuchsian-type PDE?

Non really an answer but a long comment with some (hopefully useful) suggestions. The equation you are studying seems tractable by using the method of multidimensional Mellin transform described by ...
Daniele Tampieri's user avatar

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