2.34.17 Problem 255

Internal problem ID [13915]
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, December 19, 2025 at 08:33:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.34.17.1 ✓ Maple. Time used: 0.040 (sec). Leaf size: 75

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

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

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying an equivalence, under non-integer power transformations, 
   to LODEs admitting Liouvillian solutions. 
   -> Trying a Liouvillian solution using Kovacics algorithm 
      A Liouvillian solution exists 
      Reducible group (found an exponential solution) 
      Group is reducible, not completely reducible 
      Solution has integrals. Trying a special function solution free of integr\ 
als... 
      -> Trying a solution in terms of special functions: 
         -> Bessel 
         -> elliptic 
         -> Legendre 
         -> Whittaker 
            -> hyper3: Equivalence to 1F1 under a power @ Moebius 
         -> hypergeometric 
            -> heuristic approach 
            <- heuristic approach successful 
         <- hypergeometric successful 
      <- special function solution successful 
         -> Trying to convert hypergeometric functions to elementary form... 
         <- elementary form could result into a too large expression - returnin\ 
g special function form of solution, free of uncomputed integrals 
      <- Kovacics algorithm successful 
<- Equivalence, under non-integer power transformations successful
 

2.34.17.2 ✗ 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

2.34.17.3 ✗ 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