23.5.64 problem 64

Internal problem ID [6673]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 64
Date solved : Friday, October 03, 2025 at 02:09:39 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 100

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

\[ y = \frac {\int \csc \left (x \right )^{2} \left (c_1 \left ({\mathrm e}^{-i x}+{\mathrm e}^{i x}+2\right ) \ln \left (\cos \left (\frac {x}{2}\right )^{2}\right )-i \left ({\mathrm e}^{-i x}+{\mathrm e}^{i x}+2\right ) \ln \left ({\mathrm e}^{i x}\right )+{\mathrm e}^{-i x} \left (c_1 \ln \left (2\right )-i+c_2 \right )+{\mathrm e}^{i x} \left (c_1 \ln \left (2\right )+i+c_2 \right )+2 c_1 \ln \left (2\right )+2 c_2 -\sin \left (2 x \right )\right )d x}{2}+c_3 \]

✓ Mathematica. Time used: 1.618 (sec). Leaf size: 56

ode=-D[y[x],x] + (2*Cot[x] + Csc[x])*D[y[x],{x,2}] + D[y[x],{x,3}] == Cot[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \cot \left (\frac {x}{2}\right ) \arcsin (\cos (x))-\frac {c_2 x}{\sqrt {2}}-\frac {\cot \left (\frac {x}{2}\right ) (c_2 \log (2 (\cos (x)+1))+2 c_1)}{\sqrt {2}}+c_3 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2/tan(x) + 1/sin(x))*Derivative(y(x), (x, 2)) - Derivative(y(x), x) + Derivative(y(x), (x, 3)) - 1/tan(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - Derivative(y(x), (x, 3)) - 2*Derivative(y(