23.1.114 problem 117

Internal problem ID [4721]
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 : 117
Date solved : Tuesday, September 30, 2025 at 08:19:45 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 11

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

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

✓ Mathematica. Time used: 0.097 (sec). Leaf size: 82

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

\[ \text {Solve}\left [\int _1^x(-\cos (K[1]-y(x))-\cos (K[1]+y(x)))dK[1]+\int _1^{y(x)}\left (\cos (x-K[2])-\cos (x+K[2])-\int _1^x(\sin (K[1]+K[2])-\sin (K[1]-K[2]))dK[1]\right )dK[2]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.265 (sec). Leaf size: 20

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

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