63.1.10 problem 10

Internal problem ID [15450]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 10
Date solved : Thursday, October 02, 2025 at 10:14:55 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 19

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

\[ v = -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-u}}{c_1 u}\right ) \]

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 35

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

\begin{align*} v(u)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]-1}{K[1]}dK[1]\&\right ][u+\log (u)+c_1]\\ v(u)&\to 0 \end{align*}

✓ Sympy. Time used: 0.194 (sec). Leaf size: 12

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

\[ v{\left (u \right )} = - W\left (\frac {C_{1} e^{- u}}{u}\right ) \]