Painleve ODEs - First through Sixth Transcendents
Description
Examples
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;
Painleve_ode_1≔ⅆ2ⅆx2yx=6yx2+x
Painleve_ode_2 := diff(y(x),x,x) = 2*y(x)^3+x*y(x)+a;
Painleve_ode_2≔ⅆ2ⅆx2yx=2yx3+xyx+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);
Painleve_ode_3≔ⅆ2ⅆx2yx=ⅆⅆxyx2yx−ⅆⅆxyxx+ayx2+bx+gyx3+dyx
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);
Painleve_ode_4≔ⅆ2ⅆx2yx=ⅆⅆxyx22yx+3yx32+4xyx2+2x2−ayx+byx
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);
Painleve_ode_5≔ⅆ2ⅆx2yx=12yx+1yx−1ⅆⅆxyx2−ⅆⅆxyxx+yx−12ayx+byxx2+gyxx+dyxyx+1yx−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);
Painleve_ode_6≔ⅆ2ⅆx2yx=1yx+1yx−1+1yx−xⅆⅆxyx22−1x+1x−1+1yx−xⅆⅆxyx+yxyx−1yx−xa+bxyx2+gx−1yx−12+dxx−1yx−x2x2x−12
These ODEs are irreducible. See E.L. Ince. Ordinary Differential Equations, New York: Dover Publications, 1956, 345.
All the Painleve ODEs are recognized by the odeadvisor command:
withDEtools,odeadvisor
odeadvisor
odeadvisorPainleve_ode_1
_Painleve,1st
odeadvisorPainleve_ode_2
_Painleve,2nd
odeadvisorPainleve_ode_3
_Painleve,3rd
odeadvisorPainleve_ode_4
_Painleve,4th
odeadvisorPainleve_ode_5
_Painleve,5th
odeadvisorPainleve_ode_6
_Painleve,6th
See Also
DEtools
dsolve
quadrature
missing
reducible
linear_ODEs
exact_linear
exact_nonlinear
sym_Fx
linear_sym
Bessel
Painleve
Halm
Gegenbauer
Duffing
ellipsoidal
elliptic
erf
Emden
Jacobi
Hermite
Lagerstrom
Laguerre
Liouville
Lienard
Van_der_Pol
Titchmarsh
odeadvisor,types
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