15.8.11 problem 11

Internal problem ID [3014]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 11
Date solved : Sunday, March 30, 2025 at 01:06:41 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 16

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

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

✓ Mathematica. Time used: 7.268 (sec). Leaf size: 80

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

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

✓ Sympy. Time used: 0.803 (sec). Leaf size: 17

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

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