23.4.37 problem 37

Internal problem ID [6339]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 37
Date solved : Tuesday, September 30, 2025 at 02:51:51 PM
CAS classification : [_Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 15

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

\[ y = -\arctan \left (c_1 \cot \left (x \right )-c_2 \right ) \]

✓ Mathematica. Time used: 1.508 (sec). Leaf size: 70

ode=2*Cot[x]*D[y[x],x] + 2*Tan[y[x]]*D[y[x],x]^2 + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \arctan (c_1 \cot (x)+c_2)\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2}\\ y(x)&\to -\frac {1}{2} \pi \tan (x) \sqrt {\cot ^2(x)}\\ y(x)&\to \frac {1}{2} \pi \tan (x) \sqrt {\cot ^2(x)} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE (sqrt(-2*tan(x)**2*tan(y(x))*Derivative(y(x), (x, 2)) + 1) + 1)/