2.27.8 Problem 8

Internal problem ID [13669]
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 Equations Containing Power Functions. page 213
Problem number : 8
Date solved : Friday, December 19, 2025 at 10:19:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.27.8.1 ✓ Maple. Time used: 0.030 (sec). Leaf size: 133

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

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

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
 

2.27.8.2 ✓ Mathematica. Time used: 0.229 (sec). Leaf size: 81

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

\begin{align*} y(x)&\to e^{\frac {a x^{n+1}}{n+1}} \left (c_2-\frac {c_1 2^{-\frac {1}{n+1}} x \left (\frac {a x^{n+1}}{n+1}\right )^{-\frac {1}{n+1}} \Gamma \left (\frac {1}{n+1},\frac {2 a x^{n+1}}{n+1}\right )}{n+1}\right ) \end{align*}

2.27.8.3 ✗ Sympy

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

TypeError : Mul object cannot be interpreted as an integer
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('2nd_power_series_ordinary',)