4.4.28 \(x y'(x)=a x^3 y(x) (1-x y(x))\)

ODE
\[ x y'(x)=a x^3 y(x) (1-x y(x)) \] ODE Classification

[_Bernoulli]

Book solution method
Riccati ODE, Generalized ODE

Mathematica
cpu = 0.0495087 (sec), leaf count = 76

\[\left \{\left \{y(x)\to \frac {3 e^{\frac {a x^3}{3}} \sqrt [3]{-a x^3}}{3 \sqrt [3]{-a x^3} \left (x e^{\frac {a x^3}{3}}+c_1\right )+\sqrt [3]{3} x \Gamma \left (\frac {1}{3},-\frac {a x^3}{3}\right )}\right \}\right \}\]

Maple
cpu = 0.089 (sec), leaf count = 73

\[ \left \{ \left (y \relax (x ) \right ) ^{-1}+{\frac {2\,{3}^{5/6}x\pi }{9\,\Gamma \left (2/3 \right ) }{{\rm e}^{-{\frac {a{x}^{3}}{3}}}}{\frac {1}{\sqrt [3]{-a{x}^{3}}}}}-x-{x{{\rm e}^{-{\frac {a{x}^{3}}{3}}}}\Gamma \left ({\frac {1}{3}},-{\frac {a{x}^{3}}{3}} \right ) {\frac {1}{\sqrt [3]{-9\,a{x}^{3}}}}}-{{\rm e}^{-{\frac {a{x}^{3}}{3}}}}{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x*y'[x] == a*x^3*y[x]*(1 - x*y[x]),y[x],x]

Mathematica raw output

{{y[x] -> (3*E^((a*x^3)/3)*(-(a*x^3))^(1/3))/(3*(-(a*x^3))^(1/3)*(E^((a*x^3)/3)*
x + C[1]) + 3^(1/3)*x*Gamma[1/3, -(a*x^3)/3])}}

Maple raw input

dsolve(x*diff(y(x),x) = a*x^3*(1-x*y(x))*y(x), y(x),'implicit')

Maple raw output

1/y(x)+2/9*exp(-1/3*a*x^3)*3^(5/6)*x*Pi/GAMMA(2/3)/(-a*x^3)^(1/3)-x-exp(-1/3*a*x
^3)*x*GAMMA(1/3,-1/3*a*x^3)/(-9*a*x^3)^(1/3)-exp(-1/3*a*x^3)*_C1 = 0