The general forms of the Painleve ODEs are given by the following:

Painleve_ode_1 := diff(y(x),x,x) = 6*y(x)^2+x;

$\mathrm{Painleve\_ode\_1}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=6{y\left(x\right)}^2+x$

Painleve_ode_2 := diff(y(x),x,x) = 2*y(x)^3+x*y(x)+a;

$\mathrm{Painleve\_ode\_2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=2{y\left(x\right)}^3+xy\left(x\right)+a$

Painleve_ode_3 := diff(y(x),x,x) =
diff(y(x),x)^2/y(x)-diff(y(x),x)/x+(a*y(x)^2+b)/x+g*y(x)^3+d/y(x);

$\mathrm{Painleve\_ode\_3}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{y\left(x\right)}-\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{x}+\frac{a{y\left(x\right)}^2+b}{x}+g{y\left(x\right)}^3+\frac{d}{y\left(x\right)}$

Painleve_ode_4 := diff(y(x),x,x) =
1/2*diff(y(x),x)^2/y(x)+3/2*y(x)^3+4*x*y(x)^2+2*(x^2-a)*y(x)+b/y(x);

$\mathrm{Painleve\_ode\_4}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{2y\left(x\right)}+\frac{3{y\left(x\right)}^3}{2}+4x{y\left(x\right)}^2+2\left(x^2-a\right)y\left(x\right)+\frac{b}{y\left(x\right)}$

Painleve_ode_5 := diff(y(x),x,x) =
(1/2/y(x)+1/(y(x)-1))*diff(y(x),x)^2-diff(y(x),x)/x+(y(x)-1)^2/x^2*(a*
y(x)+b/y(x))+g*y(x)/x+d*y(x)*(y(x)+1)/(y(x)-1);

$\mathrm{Painleve\_ode\_5}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\left(\frac{1}{2y\left(x\right)}+\frac{1}{y\left(x\right)-1}\right){\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2-\frac{\frac{\ⅆ}{\ⅆx}y\left(x\right)}{x}+\frac{{\left(y\left(x\right)-1\right)}^2\left(ay\left(x\right)+\frac{b}{y\left(x\right)}\right)}{x^2}+\frac{gy\left(x\right)}{x}+\frac{dy\left(x\right)\left(y\left(x\right)+1\right)}{y\left(x\right)-1}$

Painleve_ode_6 :=  diff(y(x),x,x)=1/2*(1/y(x)+1/(y(x)-1)+1/(y(x)-x))*
    diff(y(x),x)^2-(1/x+1/(x-1)+1/(y(x)-x))*diff(y(x),x)+y(x)*(y(x)-1)*
    (y(x)-x)/x^2/(x-1)^2*(a+b*x/y(x)^2+g*(x-1)/(y(x)-1)^2+d*x*(x-1)/(y(x)-x)^2);

$\mathrm{Painleve\_ode\_6}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{\left(\frac{1}{y\left(x\right)}+\frac{1}{y\left(x\right)-1}+\frac{1}{y\left(x\right)-x}\right){\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{2}-\left(\frac{1}{x}+\frac{1}{x-1}+\frac{1}{y\left(x\right)-x}\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\frac{y\left(x\right)\left(y\left(x\right)-1\right)\left(y\left(x\right)-x\right)\left(a+\frac{bx}{{y\left(x\right)}^2}+\frac{g\left(x-1\right)}{{\left(y\left(x\right)-1\right)}^2}+\frac{dx\left(x-1\right)}{{\left(y\left(x\right)-x\right)}^2}\right)}{x^2{\left(x-1\right)}^2}$

These ODEs are irreducible. See E.L. Ince. Ordinary Differential Equations, New York: Dover Publications, 1956, 345.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Painleve\_ode\_1}\right)$

$\left[\left[\mathrm{\_Painleve}\,\mathrm{1st}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{Painleve\_ode\_2}\right)$

$\left[\left[\mathrm{\_Painleve}\,\mathrm{2nd}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{Painleve\_ode\_3}\right)$

$\left[\left[\mathrm{\_Painleve}\,\mathrm{3rd}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{Painleve\_ode\_4}\right)$

$\left[\left[\mathrm{\_Painleve}\,\mathrm{4th}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{Painleve\_ode\_5}\right)$

$\left[\left[\mathrm{\_Painleve}\,\mathrm{5th}\right]\right]$

$\mathrm{odeadvisor}\left(\mathrm{Painleve\_ode\_6}\right)$

$\left[\left[\mathrm{\_Painleve}\,\mathrm{6th}\right]\right]$