Internal problem ID [8813]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1233.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-n \left (n +1\right ) y+\frac {\partial }{\partial x}\LegendreQ \left (n , x\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 420
dsolve((x^2-1)*diff(diff(y(x),x),x)-n*(n+1)*y(x)+Diff(LegendreQ(n,x),x)=0,y(x), singsol=all)
\[ y \relax (x ) = \left (-x^{2}+1\right ) \hypergeom \left (\left [1+\frac {n}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) c_{2}+\left (-x^{3}+x \right ) \hypergeom \left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) c_{1}+3 \left (n +1\right ) \left (x -1\right ) \left (x +1\right ) \left (-\hypergeom \left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \left (\int \frac {\hypergeom \left (\left [1+\frac {n}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \left (x \LegendreQ \left (n , x\right )-\LegendreQ \left (n +1, x\right )\right )}{\left (\left (-3 \hypergeom \left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )+\left (n^{2}+n -6\right ) x^{2} \hypergeom \left (\left [-\frac {n}{2}+2, \frac {n}{2}+\frac {5}{2}\right ], \left [\frac {5}{2}\right ], x^{2}\right )\right ) \hypergeom \left (\left [1+\frac {n}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )-3 \hypergeom \left (\left [\frac {n}{2}+2, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \hypergeom \left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) x^{2} \left (2+n \right ) \left (n -1\right )\right ) \left (x +1\right )^{3} \left (x -1\right )^{3}}d x \right ) x +\left (\int \frac {x \hypergeom \left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \left (x \LegendreQ \left (n , x\right )-\LegendreQ \left (n +1, x\right )\right )}{\left (\left (-3 \hypergeom \left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )+\left (n^{2}+n -6\right ) x^{2} \hypergeom \left (\left [-\frac {n}{2}+2, \frac {n}{2}+\frac {5}{2}\right ], \left [\frac {5}{2}\right ], x^{2}\right )\right ) \hypergeom \left (\left [1+\frac {n}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )-3 \hypergeom \left (\left [\frac {n}{2}+2, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \hypergeom \left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) x^{2} \left (2+n \right ) \left (n -1\right )\right ) \left (x +1\right )^{3} \left (x -1\right )^{3}}d x \right ) \hypergeom \left (\left [1+\frac {n}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )\right ) \]
✓ Solution by Mathematica
Time used: 2.427 (sec). Leaf size: 424
DSolve[(-(n*LegendreQ[-1 + n, x]) + n*x*LegendreQ[n, x])/(-1 + x^2) - n*(1 + n)*y[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \, _2F_1\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2};\frac {1}{2};x^2\right ) \left (\int _1^x\frac {3 n \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right ) K[1] (Q_{n-1}(K[1])-K[1] Q_n(K[1]))}{\left (K[1]^2-1\right )^2 \left (n \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[1]^2\right ) \left ((n+1) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[1]^2\right ) K[1]^2+3 \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )\right )-3 (n+1) \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[1]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[1]^2\right )\right )}dK[1]+c_1\right )+i x \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};x^2\right ) \left (\int _1^x\frac {3 i n \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) (Q_{n-1}(K[2])-K[2] Q_n(K[2]))}{\left (K[2]^2-1\right )^2 \left (n \, _2F_1\left (\frac {1}{2} (-n-1),\frac {n}{2};\frac {1}{2};K[2]^2\right ) \left ((n+1) \, _2F_1\left (1-\frac {n}{2},\frac {n+3}{2};\frac {5}{2};K[2]^2\right ) K[2]^2+3 \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )\right )-3 (n+1) \, _2F_1\left (\frac {1-n}{2},\frac {n}{2};\frac {1}{2};K[2]^2\right ) \, _2F_1\left (-\frac {n}{2},\frac {n+1}{2};\frac {3}{2};K[2]^2\right )\right )}dK[2]+c_2\right ) \\ \end{align*}