##### 4.11.48 $$x^2 \left (a x^3+1\right )+2 x y(x) y'(x)=6 y(x)^2$$

ODE
$x^2 \left (a x^3+1\right )+2 x y(x) y'(x)=6 y(x)^2$ ODE Classiﬁcation

[_rational, _Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.25848 (sec), leaf count = 59

$\left \{\left \{y(x)\to -\frac {1}{2} \sqrt {4 a x^5+4 c_1 x^6+x^2}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {4 a x^5+4 c_1 x^6+x^2}\right \}\right \}$

Maple
cpu = 0.024 (sec), leaf count = 45

$\left [y \left (x \right ) = -\frac {\sqrt {4 \textit {\_C1} \,x^{4}+4 a \,x^{3}+1}\, x}{2}, y \left (x \right ) = \frac {\sqrt {4 \textit {\_C1} \,x^{4}+4 a \,x^{3}+1}\, x}{2}\right ]$ Mathematica raw input

DSolve[x^2*(1 + a*x^3) + 2*x*y[x]*y'[x] == 6*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -1/2*Sqrt[x^2 + 4*a*x^5 + 4*x^6*C[1]]}, {y[x] -> Sqrt[x^2 + 4*a*x^5 +
4*x^6*C[1]]/2}}

Maple raw input

dsolve(2*x*y(x)*diff(y(x),x)+x^2*(a*x^3+1) = 6*y(x)^2, y(x))

Maple raw output

[y(x) = -1/2*(4*_C1*x^4+4*a*x^3+1)^(1/2)*x, y(x) = 1/2*(4*_C1*x^4+4*a*x^3+1)^(1/
2)*x]