2.27.7 Problem 7

Internal problem ID [13668]
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 : Sunday, January 18, 2026 at 09:05:07 PM
CAS classification : [[_Emden, _Fowler]]

2.27.7.1 second order bessel ode

0.250 (sec)

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

Entering second order bessel ode solverWriting the ode as

\begin{align*} x^{2} y^{\prime \prime }-x^{n} a \,x^{2} y = 0\tag {1} \end{align*}

Bessel ode has the form

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

The generalized form of Bessel ode is given by Bowman (1958) as the following

\begin{align*} x^{2} y^{\prime \prime }+\left (1-2 \alpha \right ) x y^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) y = 0\tag {3} \end{align*}

With the standard solution

\begin{align*} y&=x^{\alpha } \left (c_1 \operatorname {BesselJ}\left (n , \beta \,x^{\gamma }\right )+c_2 \operatorname {BesselY}\left (n , \beta \,x^{\gamma }\right )\right )\tag {4} \end{align*}

Comparing (3) to (1) and solving for \(\alpha ,\beta ,n,\gamma \) gives

\begin{align*} \alpha &= {\frac {1}{2}}\\ \beta &= \frac {2 \sqrt {-a}}{n +2}\\ n &= -\frac {1}{n +2}\\ \gamma &= 1+\frac {n}{2} \end{align*}

Substituting all the above into (4) gives the solution as

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

Summary of solutions found

\begin{align*} y &= c_1 \sqrt {x}\, \operatorname {BesselJ}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right )+c_2 \sqrt {x}\, \operatorname {BesselY}\left (-\frac {1}{n +2}, \frac {2 \sqrt {-a}\, x^{1+\frac {n}{2}}}{n +2}\right ) \\ \end{align*}

2.27.7.2 ✓ Maple. Time used: 0.036 (sec). Leaf size: 63

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

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

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 
   <- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   <- Bessel successful 
<- special function solution successful
 

2.27.7.3 ✓ 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*}

2.27.7.4 ✗ 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
 

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',)