58.4.6 problem 6

Internal problem ID [14576]
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 : Thursday, October 02, 2025 at 09:42:34 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*}

✓ Maple. Time used: 0.221 (sec). Leaf size: 29

ode:=(exp(v(u))+1)*cos(u)+exp(v(u))*(1+sin(u))*diff(v(u),u) = 0; 
dsolve(ode,v(u), singsol=all);
 

\[ v = -\ln \left (\frac {-1-\sin \left (u \right )}{-1+{\mathrm e}^{c_1} \left (1+\sin \left (u \right )\right )}\right )-c_1 \]

✓ Mathematica. Time used: 4.071 (sec). Leaf size: 37

ode=(Exp[v[u]]+1)*Cos[u] + Exp[v[u]]*(1+Sin[u])*D[ v[u],u]==0; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]
 

\begin{align*} v(u)&\to \log \left (-1+\frac {e^{c_1}}{\left (\sin \left (\frac {u}{2}\right )+\cos \left (\frac {u}{2}\right )\right )^2}\right )\\ v(u)&\to i \pi \end{align*}

✓ Sympy. Time used: 0.242 (sec). Leaf size: 12

from sympy import * 
u = symbols("u") 
v = Function("v") 
ode = Eq((exp(v(u)) + 1)*cos(u) + (sin(u) + 1)*exp(v(u))*Derivative(v(u), u),0) 
ics = {} 
dsolve(ode,func=v(u),ics=ics)
 

\[ v{\left (u \right )} = \log {\left (\frac {C_{1}}{\sin {\left (u \right )} + 1} - 1 \right )} \]