47.4.4 problem 52

Internal problem ID [7481]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number : 52
Date solved : Sunday, March 30, 2025 at 12:10:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.246 (sec). Leaf size: 49

ode:=diff(diff(y(x),x),x)-cot(x)*diff(y(x),x)+cos(x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\cos \left (x \right )+1\right ) \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \left (c_1 +c_2 \int _{}^{\cos \left (x \right )}\frac {1}{\left (\textit {\_a} +1\right )^{2} \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\textit {\_a}}{2}+\frac {1}{2}\right )^{2}}d \textit {\_a} \right ) \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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