55.34.17 problem 255

Internal problem ID [14028]
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 : 255
Date solved : Friday, October 03, 2025 at 07:23:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.104 (sec). Leaf size: 75

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

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

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -b*x**(n - 2)*(a*(n - 1) + x**n*(b - 1))*y(x) + (a + x**n)**2*Derivative(y(x), (x, 2)) cannot be solved by the hypergeometric method