35.8.10 problem 10

Internal problem ID [6217]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 13. Miscellaneous problems. page 466
Problem number : 10
Date solved : Sunday, March 30, 2025 at 10:43:29 AM
CAS classification : [_separable]

\begin{align*} u \left (1-v \right )+v^{2} \left (1-u\right ) u^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.097 (sec). Leaf size: 29

ode:=u(v)*(1-v)+v^2*(1-u(v))*diff(u(v),v) = 0; 
dsolve(ode,u(v), singsol=all);
 

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

✓ Mathematica. Time used: 3.662 (sec). Leaf size: 26

ode=u[v]*(1-v)+v^2*(1-u[v])*D[u[v],v]==0; 
ic={}; 
DSolve[{ode,ic},u[v],v,IncludeSingularSolutions->True]
 

\begin{align*} u(v)\to -W\left (v \left (-e^{\frac {1}{v}-c_1}\right )\right ) \\ u(v)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.469 (sec). Leaf size: 14

from sympy import * 
v = symbols("v") 
u = Function("u") 
ode = Eq(v**2*(1 - u(v))*Derivative(u(v), v) + (1 - v)*u(v),0) 
ics = {} 
dsolve(ode,func=u(v),ics=ics)
 

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