83.9.3 problem 3

Internal problem ID [21961]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. X at page 57
Problem number : 3
Date solved : Thursday, October 02, 2025 at 08:20:06 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} \sin \left (x \right )+\cos \left (y\right )+\cos \left (x \right )-y^{\prime } \sin \left (y\right )&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 18

ode:=sin(x)+cos(y(x))+cos(x)-diff(y(x),x)*sin(y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\pi }{2}+\arcsin \left (\sin \left (x \right )+{\mathrm e}^{-x} c_1 \right ) \]

✓ Mathematica. Time used: 11.711 (sec). Leaf size: 39

ode=(Sin[x]+Cos[y[x]] )+Cos[x]-Sin[y[x]]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\arccos \left (-\sin (x)+c_1 e^{-x}\right )\\ y(x)&\to \arccos \left (-\sin (x)+c_1 e^{-x}\right ) \end{align*}

✓ Sympy. Time used: 3.202 (sec). Leaf size: 27

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

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