69.7.19 problem 194

Internal problem ID [18099]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 194
Date solved : Thursday, October 02, 2025 at 02:42:07 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.077 (sec). Leaf size: 24

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

\[ y = -\arcsin \left ({\mathrm e}^{-x} c_1 +x +\frac {\sin \left (x \right )}{2}-\frac {\cos \left (x \right )}{2}-1\right ) \]

✓ Mathematica. Time used: 0.14 (sec). Leaf size: 67

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

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

✓ Sympy. Time used: 3.389 (sec). Leaf size: 54

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

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