23.1.125 problem 128

Internal problem ID [4732]
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 : 128
Date solved : Tuesday, September 30, 2025 at 08:20:28 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\left (1+\cos \left (x \right ) \sin \left (y\right )\right ) \tan \left (y\right ) \end{align*}

✗ Maple

ode:=diff(y(x),x) = (1+cos(x)*sin(y(x)))*tan(y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.272 (sec). Leaf size: 219

ode=D[y[x],x]==(1+Cos[x]*Sin[y[x]])*Tan[y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x-\frac {1}{4} e^{K[1]} \csc (y(x)) (2 \cos (K[1]) \csc (y(x))-\cos (K[1]-2 y(x)) \csc (y(x))-\cos (K[1]+2 y(x)) \csc (y(x))+4)dK[1]+\int _1^{y(x)}\left (e^x \cot (K[2]) \csc (K[2])-\int _1^x\left (\frac {1}{4} e^{K[1]} \cot (K[2]) \csc (K[2]) (2 \cos (K[1]) \csc (K[2])-\cos (K[1]-2 K[2]) \csc (K[2])-\cos (K[1]+2 K[2]) \csc (K[2])+4)-\frac {1}{4} e^{K[1]} \csc (K[2]) (-2 \cos (K[1]) \cot (K[2]) \csc (K[2])+\cos (K[1]-2 K[2]) \cot (K[2]) \csc (K[2])+\cos (K[1]+2 K[2]) \cot (K[2]) \csc (K[2])-2 \sin (K[1]-2 K[2]) \csc (K[2])+2 \sin (K[1]+2 K[2]) \csc (K[2]))\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 3.199 (sec). Leaf size: 53

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

\[ \left [ y{\left (x \right )} = \operatorname {asin}{\left (\frac {2 e^{x}}{C_{1} + \sqrt {2} e^{x} \sin {\left (x + \frac {\pi }{4} \right )}} \right )} + \pi , \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {2 e^{x}}{C_{1} - \sqrt {2} e^{x} \sin {\left (x + \frac {\pi }{4} \right )}} \right )}\right ] \]