54.3.203 problem 1218

Internal problem ID [12498]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1218
Date solved : Friday, October 03, 2025 at 03:19:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.066 (sec). Leaf size: 37

ode=(a + x*Cot[x])*y[x] + (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 \csc (x) \left (c_1 \operatorname {BesselJ}\left (i \sqrt {a},x\right )+c_2 \operatorname {BesselY}\left (i \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)) + (a + x/tan(x))*y(x) + (2*x**2/tan(x) + x)*Derivative(y(x), 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(x), (x, 2)) + x*y(x))/(x*(2*x + tan(x))) cannot be solved by the factorable group method