##### 4.14.29 $$\left (x^3-y(x)^3\right ) y'(x)+x^2 y(x)=0$$

ODE
$\left (x^3-y(x)^3\right ) y'(x)+x^2 y(x)=0$ ODE Classiﬁcation

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.376457 (sec), leaf count = 201

$\left \{\left \{y(x)\to \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\right \},\left \{y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\right \},\left \{y(x)\to (-1)^{2/3} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}}\right \},\left \{y(x)\to \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}}\right \},\left \{y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}}\right \},\left \{y(x)\to (-1)^{2/3} \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}}\right \}\right \}$

Maple
cpu = 0.295 (sec), leaf count = 381

$\left [y \left (x \right ) = \frac {x}{\left (\left (-\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (-\left (\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \left (\left (-\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \left (\left (-\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \left (\left (-\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \left (\left (-\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \left (-\left (\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \left (-\left (\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \left (-\left (\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}, y \left (x \right ) = \frac {x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \left (-\left (\textit {\_C1} \,x^{3}+\sqrt {\textit {\_C1}^{2} x^{6}+1}\right ) x^{3} \textit {\_C1} \right )^{\frac {1}{3}}}\right ]$ Mathematica raw input

DSolve[x^2*y[x] + (x^3 - y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve((x^3-y(x)^3)*diff(y(x),x)+x^2*y(x) = 0, y(x))

Maple raw output

[y(x) = x/((-_C1*x^3+(_C1^2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(-(_C1*x^3+(_
C1^2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(-1/2-1/2*I*3^(1/2))^2/((-_C1*x^3+(_
C1^2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(-1/2+1/2*I*3^(1/2))^2/((-_C1*x^3+(_
C1^2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(1/2-1/2*I*3^(1/2))^2/((-_C1*x^3+(_C
1^2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(1/2+1/2*I*3^(1/2))^2/((-_C1*x^3+(_C1
^2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(-1/2-1/2*I*3^(1/2))^2/(-(_C1*x^3+(_C1
^2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(-1/2+1/2*I*3^(1/2))^2/(-(_C1*x^3+(_C1
^2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(1/2-1/2*I*3^(1/2))^2/(-(_C1*x^3+(_C1^
2*x^6+1)^(1/2))*x^3*_C1)^(1/3), y(x) = x/(1/2+1/2*I*3^(1/2))^2/(-(_C1*x^3+(_C1^2
*x^6+1)^(1/2))*x^3*_C1)^(1/3)]