65.5.5 problem 10.2

Internal problem ID [13744]
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
Date solved : Tuesday, January 28, 2025 at 06:01:03 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`]]

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

✓ Solution by Maple

Time used: 0.115 (sec). Leaf size: 45

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

\[ y = \ln \left (-\frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^{2}}{\left (-4 \cos \left (\frac {x}{2}\right )^{2}+c_{1} +2 x \right ) \left (2 \cos \left (\frac {x}{2}\right )^{2}-1\right )}\right ) \]

✓ Solution by Mathematica

Time used: 6.903 (sec). Leaf size: 177

DSolve[Exp[-y[x]]*Sec[x]+2*Cos[x]-Exp[-y[x]]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^x-\frac {1}{4} e^{2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right )-y(x)} \left (e^{y(x)} \cos (2 K[1])+e^{y(x)}+1\right ) \sec (K[1])dK[1]+\int _1^{y(x)}\frac {1}{4} e^{-K[2]} \left (e^{2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )}-4 e^{K[2]} \int _1^x\left (\frac {1}{4} e^{2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right )-K[2]} \left (e^{K[2]} \cos (2 K[1])+e^{K[2]}+1\right ) \sec (K[1])-\frac {1}{4} e^{2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right )-K[2]} \left (e^{K[2]} \cos (2 K[1])+e^{K[2]}\right ) \sec (K[1])\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]