23.3.150 problem 152

Internal problem ID [5864]
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 : 152
Date solved : Friday, October 03, 2025 at 01:44:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.299 (sec). Leaf size: 69

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

\[ y = \left (c_1 \left (-1+\cos \left (x \right )\right )^{-\frac {\sqrt {-a +1}}{2}} \cos \left (\frac {x}{2}\right )^{\sqrt {-a +1}}+c_2 \left (-1+\cos \left (x \right )\right )^{\frac {\sqrt {-a +1}}{2}} \cos \left (\frac {x}{2}\right )^{-\sqrt {-a +1}}\right ) \left (\csc \left (x \right )+\cot \left (x \right )\right ) \]

✓ Mathematica. Time used: 2.103 (sec). Leaf size: 71

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

\begin{align*} y(x)&\to \left (c_2 e^{\sqrt {1-a} (\log (1-\cos (x))-\log (\cos (x)+1))}+c_1\right ) \exp \left (-\frac {1}{2} \left (\sqrt {1-a}+1\right ) (\log (1-\cos (x))-\log (\cos (x)+1))\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE (a*y(x)/sin(x) + sin(x)*Derivative(y(x), (x, 2)))/(cos(x) + 2) +