Internal problem ID [3148]
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: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {x \cos \left (y\right )^{2}+{\mathrm e}^{x} \tan \left (y\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 77
dsolve(x*cos(y(x))^2+exp(x)*tan(y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \pi -\operatorname {arccot}\left (\frac {\sqrt {2}\, \sqrt {\left (-{\mathrm e}^{x} c_{1} +x +1\right ) {\mathrm e}^{x}}}{-2 \,{\mathrm e}^{x} c_{1} +2 x +2}\right ) \\ y \left (x \right ) &= \frac {\pi }{2}-\arctan \left (\frac {\sqrt {2}\, \sqrt {\left (-{\mathrm e}^{x} c_{1} +x +1\right ) {\mathrm e}^{x}}}{-2 \,{\mathrm e}^{x} c_{1} +2 x +2}\right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 15.741 (sec). Leaf size: 149
DSolve[x*Cos[y[x]]^2+Exp[x]*Tan[y[x]]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sec ^{-1}\left (-\sqrt {2} \sqrt {e^{-x} \left (x+4 c_1 e^x+1\right )}\right ) \\ y(x)\to \sec ^{-1}\left (-\sqrt {2} \sqrt {e^{-x} \left (x+4 c_1 e^x+1\right )}\right ) \\ y(x)\to -\sec ^{-1}\left (\sqrt {2} \sqrt {e^{-x} \left (x+4 c_1 e^x+1\right )}\right ) \\ y(x)\to \sec ^{-1}\left (\sqrt {2} \sqrt {e^{-x} \left (x+4 c_1 e^x+1\right )}\right ) \\ y(x)\to -\frac {\pi }{2} \\ y(x)\to \frac {\pi }{2} \\ \end{align*}