2.30.10 Problem 119

Internal problem ID [13780]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 119
Date solved : Friday, December 19, 2025 at 12:23:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.30.10.1 ✓ Maple. Time used: 0.015 (sec). Leaf size: 130

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

\[ y = \left (a \,x^{n}-b +\frac {n}{2}+\frac {1}{2}\right ) n \,x^{-\frac {3 n}{2}+\frac {1}{2}} c_2 \operatorname {WhittakerM}\left (\frac {-2 b -n +1}{2 n}, -\frac {b}{n}+1+\frac {1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+2 x^{-\frac {3 n}{2}+\frac {1}{2}} c_2 \left (b -\frac {1}{2}-\frac {n}{2}\right )^{2} \operatorname {WhittakerM}\left (\frac {n -2 b +1}{2 n}, -\frac {b}{n}+1+\frac {1}{2 n}, \frac {2 a \,x^{n}}{n}\right )+c_1 \,x^{b} {\mathrm e}^{\frac {a \,x^{n}}{n}} \]

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 
   <- Kovacics algorithm successful 
<- Equivalence, under non-integer power transformations successful
 

2.30.10.2 ✗ Mathematica

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

Not solved

2.30.10.3 ✗ Sympy

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

ValueError : Expected Expr or iterable but got None