64.4.6 problem 6

Internal problem ID [13216]
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 : Wednesday, March 05, 2025 at 09:22:43 PM
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: 1.632 (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);
 

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

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

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+\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*}

✓ Sympy. Time used: 0.425 (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)
 

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