60.3.381 problem 1398

Internal problem ID [11377]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1398
Date solved : Sunday, March 30, 2025 at 08:18:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (3 x^{2}-1\right ) y^{\prime }}{\left (x^{2}-1\right ) x}-\frac {\left (x^{2}-1-\left (2 v +1\right )^{2}\right ) y}{\left (x^{2}-1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.139 (sec). Leaf size: 69

ode:=diff(diff(y(x),x),x) = -1/(x^2-1)*(3*x^2-1)/x*diff(y(x),x)-(x^2-1-(2*v+1)^2)/(x^2-1)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \left (x^{2}-1\right )^{-v -\frac {1}{2}} \operatorname {hypergeom}\left (\left [-v , -v \right ], \left [-2 v \right ], -x^{2}+1\right )+c_2 \left (x^{2}-1\right )^{v +\frac {1}{2}} \operatorname {hypergeom}\left (\left [v +1, v +1\right ], \left [2+2 v \right ], -x^{2}+1\right ) \]

✓ Mathematica. Time used: 0.21 (sec). Leaf size: 72

ode=D[y[x],{x,2}] == -(((-1 - (1 + 2*v)^2 + x^2)*y[x])/(-1 + x^2)^2) - ((-1 + 3*x^2)*D[y[x],x])/(x*(-1 + x^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
v = symbols("v") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (x**2 - (2*v + 1)**2 - 1)*y(x)/(x**2 - 1)**2 + (3*x**2 - 1)*Derivative(y(x), x)/(x*(x**2 - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False