23.1.667 problem 662

Internal problem ID [5274]
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 : 662
Date solved : Tuesday, September 30, 2025 at 12:06:25 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} x \left (x +6 y^{2}\right ) y^{\prime }+x y-3 y^{3}&=0 \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 38

ode:=x*(x+6*y(x)^2)*diff(y(x),x)+x*y(x)-3*y(x)^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\frac {3 c_1}{2}} \sqrt {6}}{6 x \sqrt {\frac {{\mathrm e}^{3 c_1}}{x^{3} \operatorname {LambertW}\left (\frac {6 \,{\mathrm e}^{3 c_1}}{x^{3}}\right )}}} \]

✓ Mathematica. Time used: 2.157 (sec). Leaf size: 69

ode=x(x+6 y[x]^2)D[y[x],x]+x y[x]-3 y[x]^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{3 c_1}}{x^3}\right )}}{\sqrt {6}}\\ y(x)&\to \frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{3 c_1}}{x^3}\right )}}{\sqrt {6}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 6.018 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 6*y(x)**2)*Derivative(y(x), x) + x*y(x) - 3*y(x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {e^{- 3 C_{1} - \frac {W\left (\frac {6 e^{- 6 C_{1}}}{x^{3}}\right )}{2}}}{x} \]