60.3.339 problem 1356

Internal problem ID [11335]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1356
Date solved : Sunday, March 30, 2025 at 08:17:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 29

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

\[ y = c_1 \operatorname {LegendreP}\left (v , n , \sqrt {x^{2}+1}\right )+c_2 \operatorname {LegendreQ}\left (v , n , \sqrt {x^{2}+1}\right ) \]

✓ Mathematica. Time used: 0.34 (sec). Leaf size: 78

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

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

✗ Sympy

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

ValueError : Expected Expr or iterable but got None