89.7.17 problem 17

Internal problem ID [24448]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 61
Problem number : 17
Date solved : Thursday, October 02, 2025 at 10:35:57 PM
CAS classification : [[_1st_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 28

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

\[ v = \frac {\ln \left (\tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} k -\ln \left (u^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+c_1 \right )\right ) u \right )}{2} \]

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 34

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

\[ \text {Solve}\left [\frac {1}{2} \left (\log \left (u^2+e^{4 v(u)}\right )-2 k \arctan \left (u e^{-2 v(u)}\right )\right )=c_1,v(u)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(v(u), u) - (k*exp(2*v(u)) - u)*exp(-2*v(u))/(2*(k*u +