23.3.149 problem 151

Internal problem ID [5863]
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 : 151
Date solved : Friday, October 03, 2025 at 01:44:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.115 (sec). Leaf size: 98

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

\[ y = \sqrt {\cos \left (x \right )}\, \sin \left (x \right )^{{3}/{4}} \left (\sin \left (x \right )^{-\frac {i \sqrt {23}}{4}} \operatorname {hypergeom}\left (\left [\frac {9}{8}-\frac {i \sqrt {23}}{8}, \frac {1}{8}-\frac {i \sqrt {23}}{8}\right ], \left [1-\frac {i \sqrt {23}}{4}\right ], \sin \left (x \right )^{2}\right ) c_1 +\sin \left (x \right )^{\frac {i \sqrt {23}}{4}} \operatorname {hypergeom}\left (\left [\frac {9}{8}+\frac {i \sqrt {23}}{8}, \frac {1}{8}+\frac {i \sqrt {23}}{8}\right ], \left [1+\frac {i \sqrt {23}}{4}\right ], \sin \left (x \right )^{2}\right ) c_2 \right ) \]

✓ Mathematica. Time used: 0.506 (sec). Leaf size: 146

ode=Csc[x]^2*(2 + Sin[x]^2)*y[x] - Csc[2*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 \frac {\left (-\sin ^2(x)\right )^{\frac {1}{2}-\frac {i \sqrt {23}}{8}} \cos ^2(x)^{3/8} \left (c_1 \operatorname {Hypergeometric2F1}\left (-\frac {1}{8}-\frac {i \sqrt {23}}{8},\frac {7}{8}-\frac {i \sqrt {23}}{8},\frac {3}{4},\cos ^2(x)\right )+\sqrt [4]{-1} c_2 \sqrt [4]{\cos ^2(x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{8}-\frac {i \sqrt {23}}{8},\frac {9}{8}-\frac {i \sqrt {23}}{8},\frac {5}{4},\cos ^2(x)\right )\right )}{\sqrt [8]{\sin ^2(x)} \cos ^{\frac {3}{4}}(x)} \end{align*}

✗ Sympy

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

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