4.14.11 $6xy(x{)}^2{y}^{\prime }(x)+2y(x{)}^3+x=0$

ODE
$$6xy(x{)}^2{y}^{\prime }(x)+2y(x{)}^3+x=0$$ ODE Classification

[[_homogeneous, `class G`], _exact, _rational, _Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica ✓
cpu = 0.00777719 (sec), leaf count = 99

$$\{\{y(x)\to \frac{\sqrt[3]{4c_1-x^2}}{2^{2/3}\sqrt[3]{x}}\},\{y(x)\to -\frac{\sqrt[3]{-1}\sqrt[3]{4c_1-x^2}}{2^{2/3}\sqrt[3]{x}}\},\{y(x)\to \frac{(-1{)}^{2/3}\sqrt[3]{4c_1-x^2}}{2^{2/3}\sqrt[3]{x}}\}\}$$

Maple ✓
cpu = 0.004 (sec), leaf count = 17

$$\{{\left(y\left(x\right)\right)}^3+\frac{x}{4}-\frac{\mathit{\_}\mathit{C}\mathit{1}}{x}=0\}$$ Mathematica raw input

DSolve[x + 2*y[x]^3 + 6*x*y[x]^2*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-x^2 + 4*C[1])^(1/3)/(2^(2/3)*x^(1/3))}, {y[x] -> -(((-1)^(1/3)*(-x^2
 + 4*C[1])^(1/3))/(2^(2/3)*x^(1/3)))}, {y[x] -> ((-1)^(2/3)*(-x^2 + 4*C[1])^(1/3
))/(2^(2/3)*x^(1/3))}}

Maple raw input

dsolve(6*x*y(x)^2*diff(y(x),x)+x+2*y(x)^3 = 0, y(x),'implicit')

Maple raw output

y(x)^3+1/4*x-1/x*_C1 = 0