61.33.16 problem 254

Internal problem ID [12675]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number : 254
Date solved : Friday, March 14, 2025 at 12:16:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 16.864 (sec). Leaf size: 273

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

\[ y = x^{\frac {q}{2}+\frac {1}{2}} \left (c_{1} x^{\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {HeunG}\left (-1, \frac {p q +\sqrt {q^{2}+2 q +4 s +1}\, p +p +2 r}{2 n^{2}}, \frac {\sqrt {q^{2}+2 q +4 s +1}+q +1}{2 n}, \frac {\sqrt {q^{2}+2 q +4 s +1}+q -1}{2 n}, \frac {n +\sqrt {q^{2}+2 q +4 s +1}}{n}, -\frac {-q +p}{2 n}, -a \,x^{n}\right )+c_{2} x^{-\frac {\sqrt {q^{2}+2 q +4 s +1}}{2}} \operatorname {HeunG}\left (-1, \frac {-\sqrt {q^{2}+2 q +4 s +1}\, p +\left (q +1\right ) p +2 r}{2 n^{2}}, -\frac {\sqrt {q^{2}+2 q +4 s +1}-q +1}{2 n}, -\frac {\sqrt {q^{2}+2 q +4 s +1}-q -1}{2 n}, \frac {n -\sqrt {q^{2}+2 q +4 s +1}}{n}, -\frac {-q +p}{2 n}, -a \,x^{n}\right )\right ) \]

✗ Mathematica

ode=x^2*(a^2*x^(2*n)-1)*D[y[x],{x,2}]+x*(a*p*x^n+q)*D[y[x],x]+(a*r*x^n+s)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

TypeError : Symbol object cannot be interpreted as an integer