60.3.356 problem 1373

Internal problem ID [11352]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1373
Date solved : Thursday, March 13, 2025 at 08:51:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.969 (sec). Leaf size: 84

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

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

✓ Mathematica. Time used: 0.39 (sec). Leaf size: 103

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

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

✓ Sympy. Time used: 1.457 (sec). Leaf size: 3

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

\[ y{\left (x \right )} = O\left (1\right ) \]