23.1.697 problem 692

Internal problem ID [5304]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 692
Date solved : Tuesday, September 30, 2025 at 12:25:23 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 13.148 (sec). Leaf size: 113

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

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

✓ Sympy. Time used: 0.627 (sec). Leaf size: 19

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

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