85.33.21 problem 21

Internal problem ID [22644]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 21
Date solved : Thursday, October 02, 2025 at 08:57:24 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 16

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

\[ v = {\left (\frac {1}{\operatorname {LambertW}\left (c_1 \,u^{3}\right )}\right )}^{{1}/{3}} u \]

✓ Mathematica. Time used: 5.464 (sec). Leaf size: 80

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

\begin{align*} v(u)&\to \frac {u}{\sqrt [3]{W\left (e^{-3 c_1} u^3\right )}}\\ v(u)&\to -\frac {\sqrt [3]{-1} u}{\sqrt [3]{W\left (e^{-3 c_1} u^3\right )}}\\ v(u)&\to \frac {(-1)^{2/3} u}{\sqrt [3]{W\left (e^{-3 c_1} u^3\right )}}\\ v(u)&\to 0 \end{align*}

✓ Sympy. Time used: 0.489 (sec). Leaf size: 17

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

\[ v{\left (u \right )} = e^{C_{1} + \frac {W\left (u^{3} e^{- 3 C_{1}}\right )}{3}} \]