60.3.343 problem 1360

Internal problem ID [11339]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1360
Date solved : Sunday, March 30, 2025 at 08:17:23 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*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 109

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

\[ y = \frac {c_2 \,x^{v +1} \left (x^{2}\right )^{-\frac {1}{4}-\frac {v}{2}} \Gamma \left (v +\frac {1}{2}\right )^{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -\frac {1}{2}-v , \frac {-x^{2}-1}{x^{2}-1}\right )+\pi \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x^{2}-1}{x^{2}-1}\right ) \left (x^{2}\right )^{\frac {1}{4}+\frac {v}{2}} x^{-v} c_1 \sec \left (\pi v \right )}{\sqrt {-x^{2}+1}\, \Gamma \left (v +\frac {1}{2}\right )} \]

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

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
v = symbols("v") 
y = Function("y") 
ode = Eq(-v*(v + 1)*y(x)/x**2 + 2*x*Derivative(y(x), x)/(x**2 - 1) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None