60.3.354 problem 1360

Internal problem ID [11364]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1360
Date solved : Tuesday, January 28, 2025 at 06:04:28 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}} \end{align*}

✓ Solution by Maple

Time used: 0.433 (sec). Leaf size: 109

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

\[ y = \frac {c_{2} x^{v +1} \left (x^{2}\right )^{-\frac {v}{2}-\frac {1}{4}} \Gamma \left (v +\frac {1}{2}\right )^{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -v -\frac {1}{2}, \frac {-x^{2}-1}{x^{2}-1}\right )+\sec \left (\pi v \right ) \pi \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x^{2}-1}{x^{2}-1}\right ) \left (x^{2}\right )^{\frac {1}{4}+\frac {v}{2}} x^{-v} c_{1}}{\sqrt {-x^{2}+1}\, \Gamma \left (v +\frac {1}{2}\right )} \]

✓ Solution by Mathematica

Time used: 0.151 (sec). Leaf size: 68

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

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