23.3.156 problem 158

Internal problem ID [5870]
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 : 158
Date solved : Tuesday, September 30, 2025 at 02:05:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.041 (sec). Leaf size: 38

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

\[ y = \sqrt {\sin \left (x \right )}\, \left (c_1 \sin \left (x \right )^{\frac {\sqrt {1-4 a}}{2}}+c_2 \sin \left (x \right )^{-\frac {\sqrt {1-4 a}}{2}}\right ) \]

✓ Mathematica. Time used: 2.211 (sec). Leaf size: 56

ode=a*Cot[x]^2*y[x] + Tan[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 (-\sin ^2(x)\right )^{\frac {1}{4}-\frac {1}{4} \sqrt {1-4 a}} \left (c_2 \left (-\sin ^2(x)\right )^{\frac {1}{2} \sqrt {1-4 a}}+c_1\right ) \end{align*}

✗ Sympy

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

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