2.30.3 Problem 112

Internal problem ID [13773]
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 : 112
Date solved : Thursday, January 01, 2026 at 02:35:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

2.30.3.1 second order bessel ode

0.259 (sec)

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

Entering second order bessel ode solverWriting the ode as

\begin{align*} x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-n^{2}-n \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 &= a\\ n &= -\frac {1}{2}-n\\ \gamma &= 1 \end{align*}

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

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

Summary of solutions found

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

2.30.3.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 27

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

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

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> 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
 

2.30.3.3 ✓ Mathematica. Time used: 0.022 (sec). Leaf size: 36

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

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

2.30.3.4 ✓ Sympy. Time used: 0.068 (sec). Leaf size: 41

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

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