28.1.1 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 : Monday, January 27, 2025 at 09:01:07 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*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 18

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

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

✓ Solution by Mathematica

Time used: 0.883 (sec). Leaf size: 57

DSolve[Cos[y[x]]^2+(1+Exp[-x])*Sin[y[x]]*D[y[x],x]==0,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*}