60.3.353 problem 1359

Internal problem ID [11363]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1359
Date solved : Tuesday, January 28, 2025 at 06:04:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.448 (sec). Leaf size: 57

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

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

✓ Solution by Mathematica

Time used: 0.144 (sec). Leaf size: 84

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

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