29.6.29 problem 175

Internal problem ID [4775]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 6
Problem number : 175
Date solved : Tuesday, March 04, 2025 at 07:14:51 PM
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);
 

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

✓ Mathematica. Time used: 0.232 (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: 0.586 (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)
 

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