64.4.6 problem 6

Internal problem ID [13295]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.2 (Separable equations). Exercises page 47
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 05:16:19 AM
CAS classification : [_separable]

\begin{align*} \left ({\mathrm e}^{v}+1\right ) \cos \left (u \right )+{\mathrm e}^{v} \left (1+\sin \left (u \right )\right ) v^{\prime }&=0 \end{align*}

Solution by Maple

Time used: 1.632 (sec). Leaf size: 29

dsolve((exp(v(u))+1)*cos(u) + exp(v(u))*(1+sin(u))*diff(v(u),u)=0,v(u), singsol=all)
 
\[ v \left (u \right ) = -\ln \left (\frac {-1-\sin \left (u \right )}{-1+\left (1+\sin \left (u \right )\right ) {\mathrm e}^{c_{1}}}\right )-c_{1} \]

Solution by Mathematica

Time used: 2.356 (sec). Leaf size: 64

DSolve[(Exp[v[u]]+1)*Cos[u] + Exp[v[u]]*(1+Sin[u])*D[ v[u],u]==0,v[u],u,IncludeSingularSolutions -> True]
 
\begin{align*} v(u)\to \log \left (-1+\exp \left (\int _1^u\frac {\sin \left (\frac {K[1]}{2}\right )-\cos \left (\frac {K[1]}{2}\right )}{\cos \left (\frac {K[1]}{2}\right )+\sin \left (\frac {K[1]}{2}\right )}dK[1]+c_1\right )\right ) \\ v(u)\to i \pi \\ \end{align*}