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Painleve ODEs - First through Sixth Transcendents

Description

 • 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;
 ${\mathrm{Painleve_ode_1}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}{6}{}{{y}{}\left({x}\right)}^{{2}}{+}{x}$ (1)
 > Painleve_ode_2 := diff(y(x),x,x) = 2*y(x)^3+x*y(x)+a;
 ${\mathrm{Painleve_ode_2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}{2}{}{{y}{}\left({x}\right)}^{{3}}{+}{x}{}{y}{}\left({x}\right){+}{a}$ (2)
 > 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);
 ${\mathrm{Painleve_ode_3}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)}^{{2}}}{{y}{}\left({x}\right)}{-}\frac{\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)}{{x}}{+}\frac{{a}{}{{y}{}\left({x}\right)}^{{2}}{+}{b}}{{x}}{+}{g}{}{{y}{}\left({x}\right)}^{{3}}{+}\frac{{d}}{{y}{}\left({x}\right)}$ (3)
 > 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);
 ${\mathrm{Painleve_ode_4}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}\frac{{1}}{{2}}{}\frac{{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)}^{{2}}}{{y}{}\left({x}\right)}{+}\frac{{3}}{{2}}{}{{y}{}\left({x}\right)}^{{3}}{+}{4}{}{x}{}{{y}{}\left({x}\right)}^{{2}}{+}{2}{}\left({{x}}^{{2}}{-}{a}\right){}{y}{}\left({x}\right){+}\frac{{b}}{{y}{}\left({x}\right)}$ (4)
 > 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);
 ${\mathrm{Painleve_ode_5}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}\left(\frac{{1}}{{2}{}{y}{}\left({x}\right)}{+}\frac{{1}}{{y}{}\left({x}\right){-}{1}}\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)}^{{2}}{-}\frac{\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)}{{x}}{+}\frac{{\left({y}{}\left({x}\right){-}{1}\right)}^{{2}}{}\left({a}{}{y}{}\left({x}\right){+}\frac{{b}}{{y}{}\left({x}\right)}\right)}{{{x}}^{{2}}}{+}\frac{{g}{}{y}{}\left({x}\right)}{{x}}{+}\frac{{d}{}{y}{}\left({x}\right){}\left({y}{}\left({x}\right){+}{1}\right)}{{y}{}\left({x}\right){-}{1}}$ (5)
 > 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);
 ${\mathrm{Painleve_ode_6}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}\frac{{1}}{{2}}{}\left(\frac{{1}}{{y}{}\left({x}\right)}{+}\frac{{1}}{{y}{}\left({x}\right){-}{1}}{+}\frac{{1}}{{y}{}\left({x}\right){-}{x}}\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)}^{{2}}{-}\left(\frac{{1}}{{x}}{+}\frac{{1}}{{x}{-}{1}}{+}\frac{{1}}{{y}{}\left({x}\right){-}{x}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){+}\frac{{y}{}\left({x}\right){}\left({y}{}\left({x}\right){-}{1}\right){}\left({y}{}\left({x}\right){-}{x}\right){}\left({a}{+}\frac{{b}{}{x}}{{{y}{}\left({x}\right)}^{{2}}}{+}\frac{{g}{}\left({x}{-}{1}\right)}{{\left({y}{}\left({x}\right){-}{1}\right)}^{{2}}}{+}\frac{{d}{}{x}{}\left({x}{-}{1}\right)}{{\left({y}{}\left({x}\right){-}{x}\right)}^{{2}}}\right)}{{{x}}^{{2}}{}{\left({x}{-}{1}\right)}^{{2}}}$ (6)
 These ODEs are irreducible. See E.L. Ince. Ordinary Differential Equations, New York: Dover Publications, 1956, 345.

Examples

All the Painleve ODEs are recognized by the odeadvisor command:

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (7)
 > $\mathrm{odeadvisor}\left(\mathrm{Painleve_ode_1}\right)$
 $\left[\left[{\mathrm{_Painleve}}{,}{\mathrm{1st}}\right]\right]$ (8)
 > $\mathrm{odeadvisor}\left(\mathrm{Painleve_ode_2}\right)$
 $\left[\left[{\mathrm{_Painleve}}{,}{\mathrm{2nd}}\right]\right]$ (9)
 > $\mathrm{odeadvisor}\left(\mathrm{Painleve_ode_3}\right)$
 $\left[\left[{\mathrm{_Painleve}}{,}{\mathrm{3rd}}\right]\right]$ (10)
 > $\mathrm{odeadvisor}\left(\mathrm{Painleve_ode_4}\right)$
 $\left[\left[{\mathrm{_Painleve}}{,}{\mathrm{4th}}\right]\right]$ (11)
 > $\mathrm{odeadvisor}\left(\mathrm{Painleve_ode_5}\right)$
 $\left[\left[{\mathrm{_Painleve}}{,}{\mathrm{5th}}\right]\right]$ (12)
 > $\mathrm{odeadvisor}\left(\mathrm{Painleve_ode_6}\right)$
 $\left[\left[{\mathrm{_Painleve}}{,}{\mathrm{6th}}\right]\right]$ (13)