89.6.33 problem 34

Internal problem ID [24416]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 34
Date solved : Thursday, October 02, 2025 at 10:26:28 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 62

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

\[ v = \operatorname {RootOf}\left (u \,\textit {\_Z}^{2}-c_1 +\textit {\_Z} +{\mathrm e}^{\operatorname {RootOf}\left (-u \,{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z}^{2}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+c_1 -{\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (-u \,{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z}^{2}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+c_1 -{\mathrm e}^{\textit {\_Z}}\right )} \]

✓ Mathematica. Time used: 0.201 (sec). Leaf size: 34

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

\[ \text {Solve}\left [u=\frac {e^{v(u)} (-v(u)-1)}{v(u)^2}+\frac {c_1 e^{2 v(u)}}{v(u)^2},v(u)\right ] \]

✗ Sympy

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

KeyError : ordered_hints