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Painleve ODEs - First through Sixth Transcendents

 

Description

Examples

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;

Painleve_ode_1:=ⅆ2ⅆx2yx=6yx2+x

(1)

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

Painleve_ode_2:=ⅆ2ⅆx2yx=2yx3+xyx+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);

Painleve_ode_3:=ⅆ2ⅆx2yx=ⅆⅆxyx2yxⅆⅆxyxx+ayx2+bx+gyx3+dyx

(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);

Painleve_ode_4:=ⅆ2ⅆx2yx=12ⅆⅆxyx2yx+32yx3+4xyx2+2x2ayx+byx

(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);

Painleve_ode_5:=ⅆ2ⅆx2yx=12yx+1yx1ⅆⅆxyx2ⅆⅆxyxx+yx12ayx+byxx2+gyxx+dyxyx+1yx1

(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);

Painleve_ode_6:=ⅆ2ⅆx2yx=121yx+1yx1+1yxxⅆⅆxyx21x+1x1+1yxxⅆⅆxyx+yxyx1yxxa+bxyx2+gx1yx12+dxx1yxx2x2x12

(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:

withDEtools,odeadvisor

odeadvisor

(7)

odeadvisorPainleve_ode_1

_Painleve,1st

(8)

odeadvisorPainleve_ode_2

_Painleve,2nd

(9)

odeadvisorPainleve_ode_3

_Painleve,3rd

(10)

odeadvisorPainleve_ode_4

_Painleve,4th

(11)

odeadvisorPainleve_ode_5

_Painleve,5th

(12)

odeadvisorPainleve_ode_6

_Painleve,6th

(13)

See Also

DEtools

odeadvisor

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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