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1

Following Robert's idea you can study the equation $$\ddot z +\tfrac13 \dot z-\frac{\dot z^2}{z} + \tfrac23 (z - z^2) = 0,$$ as the system $$\frac{dz}{dt}=w$$ $$\frac{dw}{dt}=\frac{w^2}{z}-\frac{w}{3}-\frac{2(z-z^2)}{3}.$$ As you said "The property I want to know is the behavior near z→∞", I would recomend you actually "see near infinity"....

0

This question gets way more difficult even for second order ODE's. Note that, by applying the Frobenius method, even if you have a Fuchsian equation you can obtain irrational exponents on some singularities. This would lead to transcendental solutions. Note also that in your examples all $a_i$'s and $c_i$'s are rational. Take for example the simple Euler ...

3

Summation defined at non-natural values is also known as "fractional summation" (even if sum limits belong to ℂ). Markus Müller and Dierk Schleicher have provided a proper axiomatic framework defining such sums, which I suggest it should be followed to make your research rigorous. Summation of convergent and divergent series under this fractional ...

11

Let $g(x) = e^{-x} f(x)$, so that $f(x) = e^x g(x)$. For a given $x$, $f'(x)$ exists if and only if $g'(x)$ exists, and $g'(x) = e^{-x} (f'(x) - f(x))$. In particular, $f'(x) = f(x)$ if and only if $g'(x) = 0$. It follows that $f$ is necessarily of the form $f(x) = e^x g(x)$, where $g$ satisfies $g'(x) = 0$ almost everywhere.

1

Substituting $y(x)=u(x)x^p$ with $p\ne0$ into your ODE, one rewrites it as $$u''-\frac{u'^2}{u}+\frac{4 u'}{3 x}+\frac{(p+2) u}{3 x^2}-\frac{2}{3} x^{p-2} u^2=0,$$ which is of the form of ODE (1) in the image of a piece of a cited paper, with $u$ in place of $y$ in ODE (1), $a=-1$, $b=4/3$, $c=(2+p)/3$, $d=-2/3$, $r=p-2\ne-2$, and $s=2\ne1$, as desired.

2

I must say that I don't much understand the motivation coming from number theory, but your question about rational solutions of ODEs has a definite answer, provided the equation has polynomial or rational coefficients. A standard reference is Abramov, S. A., Rational solutions of linear differential and difference equations with polynomial coefficients, U.S....

3

In the case that the pursuers have to actually catch the fugitive, this was answered in the article Escaping an infinitude of lions by Mikkel Abrahamsen, Jacob Holm, Eva Rotenberg, and Christian Wulff-Nilse. They prove the following much more general theorem showing that the fugitive can always escape. Theorem. For any $\epsilon >0$, a fugitive running at ...

1

The recurrence $(*)$ gives the correct form of $(**)$ as $$(1 - 4x) F'_g(x) + 2 F_g(x) = x^2 F'''_{g-1}(x) + x F''_{g-1}(x) \tag{**}$$ The boundary condition which gives the the correct form of $F_0$ is $$(1 - 4x) F'_0(x) + 2 F_0(x) = 1 \tag{***}$$ yielding $F_0 = \frac{1 + 2C_0 \sqrt{1 - 4x}}2$ and we further require $2C_0 = -1$ to give the offset Catalan ...

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