##### 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 = 0.29919 (sec), leaf count = 61

$\left \{\left \{y(x)\to \frac {e^{\frac {a x^3}{3}} \sqrt [3]{-a x^3}}{\sqrt [3]{3} x \Gamma \left (\frac {4}{3},-\frac {a x^3}{3}\right )+c_1 \sqrt [3]{-a x^3}}\right \}\right \}$

Maple
cpu = 0.086 (sec), leaf count = 126

$\left [y \left (x \right ) = -\frac {9 \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}} \left (-9 a \,x^{3}\right )^{\frac {1}{3}}}{2 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} 3^{\frac {5}{6}} x \pi \left (-9 a \,x^{3}\right )^{\frac {1}{3}}-9 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} \textit {\_C1} \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}} \left (-9 a \,x^{3}\right )^{\frac {1}{3}}-9 \,{\mathrm e}^{-\frac {a \,x^{3}}{3}} x \Gamma \left (\frac {1}{3}, -\frac {a \,x^{3}}{3}\right ) \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}}-9 x \Gamma \left (\frac {2}{3}\right ) \left (-a \,x^{3}\right )^{\frac {1}{3}} \left (-9 a \,x^{3}\right )^{\frac {1}{3}}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (E^((a*x^3)/3)*(-(a*x^3))^(1/3))/((-(a*x^3))^(1/3)*C[1] + 3^(1/3)*x*Ga
mma[4/3, -1/3*(a*x^3)])}}

Maple raw input

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

Maple raw output

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