55.30.15 problem 124

Internal problem ID [13897]
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 : 124
Date solved : Thursday, October 02, 2025 at 08:08:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

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

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

✓ Mathematica. Time used: 0.152 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.178 (sec). Leaf size: 32

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

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