55.30.4 problem 113

Internal problem ID [13886]
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 : 113
Date solved : Thursday, October 02, 2025 at 08:08:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 39

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);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 42

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},-i a x\right )+c_2 \operatorname {BesselY}\left (n+\frac {1}{2},-i a x\right )\right ) \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 54

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)
 

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