61.29.3 problem 112

Internal problem ID [12533]
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 : Monday, March 31, 2025 at 05:38:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ 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);
 

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

✓ Mathematica. Time used: 0.035 (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]
 

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

✓ Sympy. Time used: 0.118 (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)
 

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