23.3.338 problem 341

Internal problem ID [6052]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 341
Date solved : Friday, October 03, 2025 at 01:46:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

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

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

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 29

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

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

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (a*y(x)*tan(x) - x**2*tan(x)*Derivative(y(