54.3.345 problem 1362

Internal problem ID [12640]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1362
Date solved : Friday, October 03, 2025 at 03:45:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 a \,x^{2}+n \left (n +1\right ) \left (x^{2}-1\right )\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.174 (sec). Leaf size: 109

ode:=x^2*(x^2-1)*diff(diff(y(x),x),x)-2*x^3*diff(y(x),x)-((a-n)*(a+n+1)*x^2*(x^2-1)+2*x^2*a+n*(n+1)*(x^2-1))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{-n} \operatorname {HeunC}\left (0, -n -\frac {1}{2}, -2, -\frac {1}{4} a^{2}+\frac {1}{4} n^{2}-\frac {1}{4} a +\frac {1}{4} n , -\frac {1}{4} n^{2}-\frac {1}{4} n +\frac {3}{4}+\frac {1}{4} a^{2}-\frac {1}{4} a , x^{2}\right )+c_2 \,x^{n +1} \operatorname {HeunC}\left (0, n +\frac {1}{2}, -2, -\frac {1}{4} a^{2}+\frac {1}{4} n^{2}-\frac {1}{4} a +\frac {1}{4} n , -\frac {1}{4} n^{2}-\frac {1}{4} n +\frac {3}{4}+\frac {1}{4} a^{2}-\frac {1}{4} a , x^{2}\right ) \]

✗ Mathematica

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

Not solved

✗ Sympy

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

ValueError : Expected Expr or iterable but got None