Internal problem ID [11688]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 10, Two tricks for nonlinear equations. Exercises page 97
Problem number: 10.2.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]
\[ \boxed {{\mathrm e}^{-y} \sec \left (x \right )-{\mathrm e}^{-y} y^{\prime }=-2 \cos \left (x \right )} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 88
dsolve(exp(-y(x))*sec(x)+2*cos(x)-exp(-y(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \ln \left (\frac {\tan \left (\frac {x}{2}\right )^{3}+\tan \left (\frac {x}{2}\right )^{2}+\tan \left (\frac {x}{2}\right )+1}{\tan \left (\frac {x}{2}\right )^{3} c_{1} +2 \tan \left (\frac {x}{2}\right )^{3} x -\tan \left (\frac {x}{2}\right )^{2} c_{1} -2 \tan \left (\frac {x}{2}\right )^{2} x +\tan \left (\frac {x}{2}\right ) c_{1} +2 \tan \left (\frac {x}{2}\right ) x -c_{1} -2 x -4 \tan \left (\frac {x}{2}\right )+4}\right ) \]
✓ Solution by Mathematica
Time used: 2.559 (sec). Leaf size: 33
DSolve[Exp[-y[x]]*Sec[x]+2*Cos[x]-Exp[-y[x]]*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \log \left (\frac {e^{2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )}}{2 (-x+\cos (x)-2 c_1)}\right ) \]