Painlevé Transcendents

Author

Eric W. Weisstein
May 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/PainleveTranscendents.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/PainleveTranscendents.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

First

DSolve[y''[x]==6y[x]^2+x,y[x],x]

f=NDSolve[{y''[x]==6y[x]^2+x,y'[0]==0,y[0]==0},y,{x,0,10},MaxSteps->5000,WorkingPrecision->20][[1,1,-1]]

Plot[f[x],{x,0,2.6}]

Second

DSolve[y''[x]==2y[x]^3+x y[x]+α,y[x],x]

Third

DSolve[y''[x]==y'[x]^2/y[x]-y'[x]/x+(α y[x]^2+β)/x+γ y[x]^3+δ/y[x],y[x],x]

Fourth

Fifth