22.1.112 problem 135

Internal problem ID [4418]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 135
Date solved : Tuesday, September 30, 2025 at 07:31:14 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Maple. Time used: 0.088 (sec). Leaf size: 178

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

\[ y = \arctan \left (\frac {{\mathrm e}^{2 \operatorname {RootOf}\left (-x \,{\mathrm e}^{4 \textit {\_Z}}+2 c_1 \,{\mathrm e}^{3 \textit {\_Z}}+2 \,{\mathrm e}^{3 \textit {\_Z}} \textit {\_Z} -2 \,{\mathrm e}^{3 \textit {\_Z}}-2 x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} c_1 +2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}-x \right )}-1}{{\mathrm e}^{2 \operatorname {RootOf}\left (-x \,{\mathrm e}^{4 \textit {\_Z}}+2 c_1 \,{\mathrm e}^{3 \textit {\_Z}}+2 \,{\mathrm e}^{3 \textit {\_Z}} \textit {\_Z} -2 \,{\mathrm e}^{3 \textit {\_Z}}-2 x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} c_1 +2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}-x \right )}+1}, \operatorname {sech}\left (\operatorname {RootOf}\left (-x \,{\mathrm e}^{4 \textit {\_Z}}+2 c_1 \,{\mathrm e}^{3 \textit {\_Z}}+2 \,{\mathrm e}^{3 \textit {\_Z}} \textit {\_Z} -2 \,{\mathrm e}^{3 \textit {\_Z}}-2 x \,{\mathrm e}^{2 \textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} c_1 +2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}}-x \right )\right )\right ) \]

✓ Mathematica. Time used: 0.276 (sec). Leaf size: 26

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

\[ \text {Solve}[x=\cos (y(x)) (\text {arctanh}(\sin (y(x)))-\sin (y(x)))+c_1 \cos (y(x)),y(x)] \]

✗ Sympy

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

Timed Out