Internal problem ID [9561]
Internal file name [OUTPUT/8501_Monday_June_06_2022_03_15_08_AM_51751516/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1232.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _linear, _nonhomogeneous]]
Unable to solve or complete the solution.
\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-n \left (n +1\right ) y=-\frac {\left (n +1\right ) \operatorname {LegendreP}\left (n +1, x\right )-\left (n +1\right ) x \operatorname {LegendreP}\left (n , x\right )}{x^{2}-1}} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] -> Try solving first the homogeneous part of the ODE checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius -> hypergeometric -> heuristic approach -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 2F1 ODE <- hypergeometric successful <- special function solution successful <- solving first the homogeneous part of the ODE successful`
✓ Solution by Maple
Time used: 0.125 (sec). Leaf size: 409
dsolve((x^2-1)*diff(diff(y(x),x),x)-n*(n+1)*y(x)+Diff(LegendreP(n,x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = 3 \left (x +1\right ) \left (x -1\right ) \left (-\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \left (1+n \right ) \left (\int \frac {x \operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \left (x \operatorname {LegendreP}\left (n , x\right )-\operatorname {LegendreP}\left (1+n , x\right )\right )}{3 \left (x +1\right )^{3} \left (x -1\right )^{3} \left (\left (\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )+\left (n^{2}+n -2\right ) x^{2} \operatorname {hypergeom}\left (\left [\frac {n}{2}+2, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )-\frac {\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+2, \frac {n}{2}+\frac {5}{2}\right ], \left [\frac {5}{2}\right ], x^{2}\right ) x^{2} \left (n +3\right ) \left (n -2\right )}{3}\right )}d x \right )+\operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) x \left (1+n \right ) \left (\int \frac {\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \left (x \operatorname {LegendreP}\left (n , x\right )-\operatorname {LegendreP}\left (1+n , x\right )\right )}{3 \left (x +1\right )^{3} \left (x -1\right )^{3} \left (\left (\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )+\left (n^{2}+n -2\right ) x^{2} \operatorname {hypergeom}\left (\left [\frac {n}{2}+2, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )-\frac {\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+2, \frac {n}{2}+\frac {5}{2}\right ], \left [\frac {5}{2}\right ], x^{2}\right ) x^{2} \left (n +3\right ) \left (n -2\right )}{3}\right )}d x \right )-\frac {\operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) c_{1} x}{3}-\frac {\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) c_{2}}{3}\right ) \]
✓ Solution by Mathematica
Time used: 8.981 (sec). Leaf size: 468
DSolve[(-(n*LegendreP[-1 + n, x]) + n*x*LegendreP[n, x])/(-1 + x^2) - n*(1 + n)*y[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2},\frac {1}{2},x^2\right ) \int _1^x\frac {3 n \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[1]^2\right ) K[1] (\operatorname {LegendreP}(n-1,K[1])-K[1] \operatorname {LegendreP}(n,K[1]))}{\left (K[1]^2-1\right )^2 \left (n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-1),\frac {n}{2},\frac {1}{2},K[1]^2\right ) \left ((n+1) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},\frac {n+3}{2},\frac {5}{2},K[1]^2\right ) K[1]^2+3 \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[1]^2\right )\right )-3 (n+1) \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {n}{2},\frac {1}{2},K[1]^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[1]^2\right )\right )}dK[1]+i x \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},x^2\right ) \int _1^x\frac {3 i n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-1),\frac {n}{2},\frac {1}{2},K[2]^2\right ) (\operatorname {LegendreP}(n-1,K[2])-K[2] \operatorname {LegendreP}(n,K[2]))}{\left (K[2]^2-1\right )^2 \left (n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-n-1),\frac {n}{2},\frac {1}{2},K[2]^2\right ) \left ((n+1) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},\frac {n+3}{2},\frac {5}{2},K[2]^2\right ) K[2]^2+3 \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[2]^2\right )\right )-3 (n+1) \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},\frac {n}{2},\frac {1}{2},K[2]^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[2]^2\right )\right )}dK[2]+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2},\frac {1}{2},x^2\right )+i c_2 x \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},x^2\right ) \]