23.1.178 problem 178

Internal problem ID [4785]
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 : 178
Date solved : Tuesday, September 30, 2025 at 08:36:05 AM
CAS classification : [_Bernoulli]

\begin{align*} x y^{\prime }&=a \,x^{3} \left (1-x y\right ) y \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 95

ode:=x*diff(y(x),x) = a*x^3*(1-x*y(x))*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {3 \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{{1}/{3}} 3^{{2}/{3}}}{3 \Gamma \left (\frac {2}{3}\right ) 3^{{2}/{3}} {\mathrm e}^{-\frac {a \,x^{3}}{3}} c_1 \left (-a \,x^{3}\right )^{{1}/{3}}+3 \Gamma \left (\frac {2}{3}\right ) 3^{{2}/{3}} \left (-a \,x^{3}\right )^{{1}/{3}} x +3 \Gamma \left (\frac {2}{3}\right ) {\mathrm e}^{-\frac {a \,x^{3}}{3}} \Gamma \left (\frac {1}{3}, -\frac {a \,x^{3}}{3}\right ) x -2 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} \sqrt {3}\, \pi x} \]

✓ Mathematica. Time used: 0.148 (sec). Leaf size: 66

ode=x*D[y[x],x]==a*x^3*(1-x*y[x])*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{\frac {a x^3}{3}} \sqrt [3]{-a x^3}}{\sqrt [3]{3} x \Gamma \left (\frac {4}{3},-\frac {a x^3}{3}\right )+c_1 \sqrt [3]{-a x^3}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.352 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*x**3*(-x*y(x) + 1)*y(x) + x*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\sqrt [3]{a} e^{\frac {a x^{3}}{3}}}{C_{1} \sqrt [3]{a} + \left (-1\right )^{\frac {2}{3}} \sqrt [3]{3} \gamma \left (\frac {4}{3}, \frac {a x^{3} e^{i \pi }}{3}\right )} \]