22.1.1 problem 1

Internal problem ID [4307]
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 : 1
Date solved : Tuesday, September 30, 2025 at 07:16:58 AM
CAS classification : [_separable]

\begin{align*} \cos \left (y\right )^{2}+\left (1+{\mathrm e}^{-x}\right ) \sin \left (y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 21

ode:=cos(y(x))^2+(1+exp(-x))*sin(y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.534 (sec). Leaf size: 57

ode=Cos[y[x]]^2+(1+Exp[-x])*Sin[y[x]]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sec ^{-1}\left (-\log \left (e^x+1\right )+2 c_1\right )\\ y(x)&\to \sec ^{-1}\left (-\log \left (e^x+1\right )+2 c_1\right )\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 0.362 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 + exp(-x))*sin(y(x))*Derivative(y(x), x) + cos(y(x))**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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