4.14.36 $2y(x{)}^3{y}^{\prime }(x)=x^3-xy(x{)}^2$

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

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.101282 (sec), leaf count = 878

$$\{\{y(x)\to -\frac{\sqrt{\frac{x^4}{\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}}-x^2+\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}}}{\sqrt{2}}\},\{y(x)\to \frac{\sqrt{\frac{x^4}{\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}}-x^2+\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}}}{\sqrt{2}}\},\{y(x)\to -\frac{1}{2}\sqrt{\frac{i(i+\sqrt{3})x^4-2\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}{x}^2+(-1-i\sqrt{3})(x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}){}^{2/3}}{\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}}}\},\{y(x)\to \frac{1}{2}\sqrt{\frac{i(i+\sqrt{3})x^4-2\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}{x}^2+(-1-i\sqrt{3})(x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}){}^{2/3}}{\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}}}\},\{y(x)\to -\frac{1}{2}\sqrt{\frac{(-1-i\sqrt{3})x^4-2\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}{x}^2+i(i+\sqrt{3})(x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}){}^{2/3}}{\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}}}\},\{y(x)\to \frac{1}{2}\sqrt{\frac{(-1-i\sqrt{3})x^4-2\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}{x}^2+i(i+\sqrt{3})(x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}){}^{2/3}}{\sqrt[3]{x^6-2e^{12c_1}+2\sqrt{e^{24c_1}-e^{12c_1}{x}^6}}}}\}\}$$

Maple ✓
cpu = 0.023 (sec), leaf count = 45

$$\{-\frac{1}{6}\ln \left(\frac{-x^2+2{\left(y\left(x\right)\right)}^2}{x^2}\right)-\frac{1}{3}\ln \left(\frac{x^2+{\left(y\left(x\right)\right)}^2}{x^2}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[2*y[x]^3*y'[x] == x^3 - x*y[x]^2,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-x^2 + x^4/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C
[1])*x^6])^(1/3) + (-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6]
)^(1/3)]/Sqrt[2])}, {y[x] -> Sqrt[-x^2 + x^4/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(2
4*C[1]) - E^(12*C[1])*x^6])^(1/3) + (-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) -
 E^(12*C[1])*x^6])^(1/3)]/Sqrt[2]}, {y[x] -> -Sqrt[(I*(I + Sqrt[3])*x^4 - 2*x^2*
(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + (-1 - I*S
qrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(2/3))/(-
2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]/2}, {y[x] ->
 Sqrt[(I*(I + Sqrt[3])*x^4 - 2*x^2*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - 
E^(12*C[1])*x^6])^(1/3) + (-1 - I*Sqrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*
C[1]) - E^(12*C[1])*x^6])^(2/3))/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^
(12*C[1])*x^6])^(1/3)]/2}, {y[x] -> -Sqrt[((-1 - I*Sqrt[3])*x^4 - 2*x^2*(-2*E^(1
2*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3) + I*(I + Sqrt[3])*(
-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(2/3))/(-2*E^(12*C
[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])*x^6])^(1/3)]/2}, {y[x] -> Sqrt[((-
1 - I*Sqrt[3])*x^4 - 2*x^2*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[
1])*x^6])^(1/3) + I*(I + Sqrt[3])*(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E
^(12*C[1])*x^6])^(2/3))/(-2*E^(12*C[1]) + x^6 + 2*Sqrt[E^(24*C[1]) - E^(12*C[1])
*x^6])^(1/3)]/2}}

Maple raw input

dsolve(2*y(x)^3*diff(y(x),x) = x^3-x*y(x)^2, y(x),'implicit')

Maple raw output

-1/6*ln((-x^2+2*y(x)^2)/x^2)-1/3*ln((x^2+y(x)^2)/x^2)-ln(x)-_C1 = 0