23.3.337 problem 340

Internal problem ID [6051]
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 : 340
Date solved : Friday, October 03, 2025 at 01:45:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 37

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

\[ y = \sqrt {x}\, \csc \left (x \right ) \left (c_1 \operatorname {BesselJ}\left (\frac {\sqrt {-4 a +1}}{2}, x\right )+c_2 \operatorname {BesselY}\left (\frac {\sqrt {-4 a +1}}{2}, x\right )\right ) \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 50

ode=a*y[x] + 2*x^2*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 \sqrt {x} \csc (x) \left (c_1 \operatorname {BesselJ}\left (\frac {1}{2} \sqrt {1-4 a},x\right )+c_2 \operatorname {BesselY}\left (\frac {1}{2} \sqrt {1-4 a},x\right )\right ) \end{align*}

✗ Sympy

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

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