23.1.116 problem 119

Internal problem ID [4723]
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 : 119
Date solved : Tuesday, September 30, 2025 at 08:19:51 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 14

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

\[ y = \arccos \left ({\mathrm e}^{-\cos \left (x \right )} c_1 +1\right ) \]

✓ Mathematica. Time used: 0.248 (sec). Leaf size: 110

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

\[ \text {Solve}\left [\int _1^x-e^{\text {arctanh}(\cos (y(x)))} \left (\cos \left (K[1]-\frac {y(x)}{2}\right )-\cos \left (K[1]+\frac {y(x)}{2}\right )\right ) \sec \left (\frac {y(x)}{2}\right )dK[1]-y(x) \int _1^x0dK[1]+\sqrt {\sin ^2(y(x))} \left (-\csc \left (\frac {y(x)}{2}\right )\right ) \sec \left (\frac {y(x)}{2}\right ) \left (\log \left (\sec ^2\left (\frac {y(x)}{2}\right )\right )-2 \log \left (\tan \left (\frac {y(x)}{2}\right )\right )\right )=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.467 (sec). Leaf size: 20

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

\[ \int \limits ^{y{\left (x \right )}} \frac {\sin {\left (y \right )} \tan {\left (y \right )}}{\sin {\left (y \right )} - \tan {\left (y \right )}}\, dy = C_{1} + \cos {\left (x \right )} \]