23.1.117 problem 120

Internal problem ID [4724]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 120
Date solved : Tuesday, September 30, 2025 at 08:19:54 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 11

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

\[ y = \arccos \left (\frac {\cos \left (x \right )}{c_1}\right ) \]

✓ Mathematica. Time used: 4.628 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to -\arccos \left (\frac {1}{2} c_1 \cos (x)\right )\\ y(x)&\to \arccos \left (\frac {1}{2} c_1 \cos (x)\right )\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 0.246 (sec). Leaf size: 20

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

\[ \left [ y{\left (x \right )} = - \operatorname {acos}{\left (C_{1} \cos {\left (x \right )} \right )} + 2 \pi , \ y{\left (x \right )} = \operatorname {acos}{\left (C_{1} \cos {\left (x \right )} \right )}\right ] \]