2.29.6 Problem 66

Internal problem ID [13727]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 66
Date solved : Friday, December 19, 2025 at 11:18:53 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+\left (1-3 n \right ) y^{\prime }-a^{2} n^{2} x^{2 n -1} y&=0 \\ \end{align*}

2.29.6.1 ✓ Maple. Time used: 0.019 (sec). Leaf size: 62

ode:=x*diff(diff(y(x),x),x)+(1-3*n)*diff(y(x),x)-a^2*n^2*x^(2*n-1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -{\mathrm e}^{x^{n} a} c_1 \left (x^{n} a -x^{-n} \sqrt {x^{2 n}}\right )+c_2 \,{\mathrm e}^{-x^{n} a} \left (x^{n} a +x^{-n} \sqrt {x^{2 n}}\right ) \]

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 
      Group is reducible or imprimitive 
   <- Kovacics algorithm successful 
<- Equivalence, under non-integer power transformations successful
 

2.29.6.2 ✓ Mathematica. Time used: 0.049 (sec). Leaf size: 77

ode=x*D[y[x],{x,2}]+(1-3*n)*D[y[x],x]-a^2*n^2*x^(2*n-1)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (c_1-\frac {3}{8} i a c_2 \sqrt {x^{2 n}}\right ) \cosh \left (a \sqrt {x^{2 n}}\right )+\frac {1}{8} \left (3 i c_2-8 a c_1 \sqrt {x^{2 n}}\right ) \sinh \left (a \sqrt {x^{2 n}}\right ) \end{align*}

2.29.6.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-a**2*n**2*x**(2*n - 1)*y(x) + x*Derivative(y(x), (x, 2)) + (1 - 3*n)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None