23.3.143 problem 145

Internal problem ID [5857]
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 : 145
Date solved : Friday, October 03, 2025 at 01:44:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.176 (sec). Leaf size: 29

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

\[ y = c_1 \cos \left (2 x \right )+c_2 \sin \left (2 x \right )^{{3}/{2}} \operatorname {hypergeom}\left (\left [\frac {1}{2}, 2\right ], \left [\frac {7}{4}\right ], \sin \left (x \right )^{2}\right ) \]

✓ Mathematica. Time used: 20.148 (sec). Leaf size: 64

ode=2*y[x] - Cot[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 {2}{3} c_2 \cos (2 x) \cos ^{\frac {3}{2}}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{4},\frac {7}{4},\cos ^2(x)\right )+\frac {1}{2} c_1 \cos (2 x)-2 c_2 \sin ^2(x)^{3/4} \cos ^{\frac {3}{2}}(x) \end{align*}

✗ Sympy

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

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