60.3.227 problem 1231

Internal problem ID [11237]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1231
Date solved : Tuesday, January 28, 2025 at 05:42:12 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }-v \left (v +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.175 (sec). Leaf size: 52

dsolve((x^2-1)*diff(diff(y(x),x),x)-v*(v+1)*y(x)=0,y(x), singsol=all)
 

\[ y = -\left (x -1\right ) \left (x +1\right ) \left (\operatorname {hypergeom}\left (\left [1-\frac {v}{2}, \frac {3}{2}+\frac {v}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) c_{2} x +c_{1} \operatorname {hypergeom}\left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.101 (sec). Leaf size: 56

DSolve[-(v*(1 + v)*y[x]) + (-1 + x^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (-\frac {v}{2}-\frac {1}{2},\frac {v}{2},\frac {1}{2},x^2\right )+i c_2 x \operatorname {Hypergeometric2F1}\left (-\frac {v}{2},\frac {v+1}{2},\frac {3}{2},x^2\right ) \]