58.2.20 problem 21

Internal problem ID [9143]
Book : Second order enumerated odes
Section : section 2
Problem number : 21
Date solved : Sunday, March 30, 2025 at 02:23:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }+\cot \left (x \right ) y^{\prime }+4 y \csc \left (x \right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)+cot(x)*diff(y(x),x)+4*y(x)*csc(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \left (\csc \left (x \right )+\cot \left (x \right )\right )^{-2 i}+c_2 \left (\csc \left (x \right )+\cot \left (x \right )\right )^{2 i} \]

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 25

ode=D[y[x],{x,2}]+Cot[x]*D[y[x],x]+4*y[x]*Csc[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos (2 \text {arctanh}(\cos (x)))-c_2 \sin (2 \text {arctanh}(\cos (x))) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x)/sin(x)**2 + Derivative(y(x), (x, 2)) + Derivative(y(x), x)/tan(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE 4*y(x)*tan(x)/sin(x)**2 + tan(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x) cannot be solved by the factorable group method