2.30.9 Problem 118

Internal problem ID [13779]
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 : 118
Date solved : Sunday, January 18, 2026 at 09:15:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

2.30.9.1 second order bessel ode

0.185 (sec)

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

Entering second order bessel ode solverWriting the ode as

\begin{align*} x^{2} y^{\prime \prime }+\left (x^{n} a +b \right ) 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}\\ n &= \frac {\sqrt {-4 b +1}}{n}\\ \gamma &= \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 {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right )+c_2 \sqrt {x}\, \operatorname {BesselY}\left (\frac {\sqrt {-4 b +1}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) \end{align*}

Summary of solutions found

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

2.30.9.2 ✓ Maple. Time used: 0.007 (sec). Leaf size: 63

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

\[ y = \left (\operatorname {BesselY}\left (\frac {\sqrt {1-4 b}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) c_2 +\operatorname {BesselJ}\left (\frac {\sqrt {1-4 b}}{n}, \frac {2 \sqrt {a}\, x^{\frac {n}{2}}}{n}\right ) c_1 \right ) \sqrt {x} \]

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

2.30.9.3 ✓ Mathematica. Time used: 0.103 (sec). Leaf size: 351

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

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

2.30.9.4 ✓ Sympy. Time used: 0.328 (sec). Leaf size: 61

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*x**n + b)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {2 \sqrt {\frac {1}{4} - b}}{n}}\left (\frac {2 \sqrt {a} x^{\frac {n}{2}}}{n}\right ) + C_{2} Y_{\frac {2 \sqrt {\frac {1}{4} - b}}{n}}\left (\frac {2 \sqrt {a} x^{\frac {n}{2}}}{n}\right )\right ) \]

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_linear_bessel', '2nd_power_series_regular')