The canonical form of the first Painlevé equation The first Painlevé equation is traditionally written as
$$y''=6y^2+x. $$
Using scaling in both the dependent and independent variables, one can transform this equation into
$$Y''=aY^2+bx $$
for arbitrary complex constants $a,b$. My question is: is there any special reason for choosing the coefficients $a=6,b=1$ in the traditional form? If so, what is it?
My guess regarding the $a=6$ part is that this is the only choice which makes the principal part of the Laurent series expansion around any pole $x_0$ have the form
$$\frac{1}{(x-x_0)^2},$$
shared by the solutions of the Weierstrasß differential equation
$$\wp''=6\wp^2-\frac{1}{2}g_2. $$
However, I can't see any reason for preferring $b=1$ over any other value.
 A: The parameter $b$ would enter further along in the Laurent series at a pole. Specifically, for the equation $y'' = 6y^2 + bx$, this expansion is
$$\frac{1}{(x-x_0)^2} - \frac{b x_0}{10} (x-x_0)^2 - \frac{1}{6} (x-x_0)^3 + O\left( (x-x_0)^4 \right).$$
Since the next to leading coefficient depends on the position $x_0$ of the movable pole, one cannot use $b$ to scale away some factor in the expansion, as this would only hold at one pole. So, the plain answer is, there is no real reason to set $b=1$, and other conventions are sometimes found, such as $b=6$ or $b=-1$.
Among all equations of the form
$$y'' = 6y^2 + f(x),$$
only those obeying $f''(x) \equiv 0$ have the property that all their movable singularities are poles (Painlevé property), i.e. when $f(x) = bx + c$. It seems natural to pick $b=1, c=0$ as the canonical representative in this class, to which it can be re-scaled, unless $b=0$, in which case it is reduced to an equation with elliptic solutions.
A: You can find Painlevé's derivation in his paper: Mémoire sur les équations différentielles dont l’intégrale générale est uniforme Bulletin de la S. M. F., tome 28 (1900), p. 201-261.
In particular, section 14 of that paper shows how he arrives at what we now call the first Painlevé transcendent.
Chapter 14 of Ince's book, Ordinary Differential Equations (public domain) gives an account in English.
