23.3.152 problem 154

Internal problem ID [5866]
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 : 154
Date solved : Tuesday, September 30, 2025 at 02:05:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.231 (sec). Leaf size: 50

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

\[ y = {\mathrm e}^{\frac {\cos \left (x \right ) \operatorname {csgn}\left (\csc \left (x \right )\right )}{2}} \left (c_1 \sinh \left (\frac {\sqrt {3}\, \cot \left (x \right )}{2 \sqrt {-\csc \left (x \right )^{2}}}\right )+c_2 \cosh \left (\frac {\sqrt {3}\, \cot \left (x \right )}{2 \sqrt {-\csc \left (x \right )^{2}}}\right )\right ) \]

✓ Mathematica. Time used: 0.099 (sec). Leaf size: 45

ode=Sin[x]^2*y[x] - (Cot[x] - Sin[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 e^{\frac {\cos (x)}{2}} \left (c_1 \cos \left (\frac {1}{2} \sqrt {3} \cos (x)\right )+c_2 \sin \left (\frac {1}{2} \sqrt {3} \cos (x)\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(-sin(x) + 1/tan(x))*Derivative(y(x), x) + y(x)*sin(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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