54.3.385 problem 1402

Internal problem ID [12680]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1402
Date solved : Friday, October 03, 2025 at 03:45:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (\left (1-4 a \right ) x^{2}-1\right ) y^{\prime }}{x \left (x^{2}-1\right )}-\frac {\left (\left (-v^{2}+x^{2}\right ) \left (x^{2}-1\right )^{2}+4 a \left (a +1\right ) x^{4}-2 a \,x^{2} \left (x^{2}-1\right )\right ) y}{x^{2} \left (x^{2}-1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.192 (sec). Leaf size: 56

ode:=diff(diff(y(x),x),x) = -1/x/(x^2-1)*((1-4*a)*x^2-1)*diff(y(x),x)-((-v^2+x^2)*(x^2-1)^2+4*a*(a+1)*x^4-2*a*x^2*(x^2-1))/x^2/(x^2-1)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✗ Mathematica

ode=D[y[x],{x,2}] == -(((4*a*(1 + a)*x^4 - 2*a*x^2*(-1 + x^2) + (-1 + x^2)^2*(-v^2 + x^2))*y[x])/(x^2*(-1 + x^2)^2)) - ((-1 + (1 - 4*a)*x^2)*D[y[x],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") 
v = symbols("v") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + (x**2*(1 - 4*a) - 1)*Derivative(y(x), x)/(x*(x**2 - 1)) + (4*a*x**4*(a + 1) - 2*a*x**2*(x**2 - 1) + (-v**2 + x**2)*(x**2 - 1)**2)*y(x)/(x**2*(x**2 - 1)**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None