New answers tagged differential-equations
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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{...
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} \...
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 ...
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 ...
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