55.27.7 problem 7

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 Equations Containing Power Functions. page 213
Problem number : 7
Date solved : Thursday, October 02, 2025 at 08:06:39 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }-a \,x^{n} y&=0 \end{align*}

✓ Maple. Time used: 0.074 (sec). Leaf size: 63

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

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

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 119

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

\begin{align*} y(x)&\to (n+2)^{-\frac {1}{n+2}} \sqrt {x} a^{\frac {1}{2 n+4}} \left (c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselI}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right )+c_2 (-1)^{\frac {1}{n+2}} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselI}\left (\frac {1}{n+2},\frac {2 \sqrt {a} x^{\frac {n}{2}+1}}{n+2}\right )\right ) \end{align*}

✗ Sympy

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

TypeError : Symbol object cannot be interpreted as an integer