85.33.50 problem 50

Internal problem ID [22673]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 50
Date solved : Thursday, October 02, 2025 at 09:05:22 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 16

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

\[ \ln \left (x \right )-\ln \left (y+\sin \left (y\right )\right )+c_1 = 0 \]

✓ Mathematica. Time used: 0.331 (sec). Leaf size: 19

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

\begin{align*} y(x)&\to \text {InverseFunction}[\log (\text {$\#$1}+\sin (\text {$\#$1}))\&][\log (x)+c_1] \end{align*}

✓ Sympy. Time used: 1.428 (sec). Leaf size: 14

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

\[ \log {\left (y{\left (x \right )} + \sin {\left (y{\left (x \right )} \right )} \right )} = C_{1} + \log {\left (x \right )} \]