58.4.10 problem 10

Internal problem ID [14580]
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 : 10
Date solved : Thursday, October 02, 2025 at 09:42:45 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} v^{3}+\left (u^{3}-u v^{2}\right ) v^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 32

ode:=v(u)^3+(u^3-u*v(u)^2)*diff(v(u),u) = 0; 
dsolve(ode,v(u), singsol=all);
 

\[ v = \frac {{\mathrm e}^{-c_1}}{\sqrt {-\frac {{\mathrm e}^{-2 c_1}}{u^{2} \operatorname {LambertW}\left (-\frac {{\mathrm e}^{-2 c_1}}{u^{2}}\right )}}} \]

✓ Mathematica. Time used: 2.499 (sec). Leaf size: 56

ode=v[u]^3+ (u^3-u*v[u]^2)*D[ v[u],u]==0; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]
 

\begin{align*} v(u)&\to -i u \sqrt {W\left (-\frac {e^{-2 c_1}}{u^2}\right )}\\ v(u)&\to i u \sqrt {W\left (-\frac {e^{-2 c_1}}{u^2}\right )}\\ v(u)&\to 0 \end{align*}

✓ Sympy. Time used: 0.892 (sec). Leaf size: 19

from sympy import * 
u = symbols("u") 
v = Function("v") 
ode = Eq((u**3 - u*v(u)**2)*Derivative(v(u), u) + v(u)**3,0) 
ics = {} 
dsolve(ode,func=v(u),ics=ics)
 

\[ v{\left (u \right )} = e^{C_{1} - \frac {W\left (- \frac {e^{2 C_{1}}}{u^{2}}\right )}{2}} \]