##### 4.15.10 $$\left (x^3-y(x)^4\right ) y'(x)=3 x^2 y(x)$$

ODE
$\left (x^3-y(x)^4\right ) y'(x)=3 x^2 y(x)$ ODE Classiﬁcation

[[_homogeneous, class G], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.402734 (sec), leaf count = 1021

$\left \{\left \{y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}}-\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}-\frac {6 c_1}{\sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}}+\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}-\frac {6 c_1}{\sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}}}}\right \},\left \{y(x)\to -\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}}-\frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}+\frac {6 c_1}{\sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}}}}\right \},\left \{y(x)\to \frac {1}{2} \sqrt {-\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}+\frac {6 c_1}{\sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}}}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x^3}{\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}+\frac {\sqrt [3]{9 c_1{}^2-\sqrt {81 c_1{}^4-256 x^9}}}{\sqrt [3]{2}}}\right \}\right \}$

Maple
cpu = 0.431 (sec), leaf count = 25

$\left [y \left (x \right ) = \RootOf \left (x^{9} \textit {\_Z}^{4}+3-{\mathrm e}^{\frac {9 \textit {\_C1}}{4}} \textit {\_Z} \right ) x^{3}\right ]$ Mathematica raw input

DSolve[(x^3 - y[x]^4)*y'[x] == 3*x^2*y[x],y[x],x]

Mathematica raw output

{{y[x] -> Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) + (
9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]/2 - Sqrt[(-4*2^(1/3)*x^3)/
(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) - (9*C[1]^2 - Sqrt[-256*x^9 + 81*C
[1]^4])^(1/3)/2^(1/3) - (6*C[1])/Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9
+ 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]]/2}
, {y[x] -> Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) +
(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]/2 + Sqrt[(-4*2^(1/3)*x^3)
/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) - (9*C[1]^2 - Sqrt[-256*x^9 + 81*
C[1]^4])^(1/3)/2^(1/3) - (6*C[1])/Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9
 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)]]/2
}, {y[x] -> -1/2*Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1
/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)] - Sqrt[(-4*2^(1/3)*
x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) - (9*C[1]^2 - Sqrt[-256*x^9 +
 81*C[1]^4])^(1/3)/2^(1/3) + (6*C[1])/Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256
*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)
]]/2}, {y[x] -> -1/2*Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4]
)^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(1/3)] + Sqrt[(-4*2^(1
/3)*x^3)/(9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3) - (9*C[1]^2 - Sqrt[-256*x
^9 + 81*C[1]^4])^(1/3)/2^(1/3) + (6*C[1])/Sqrt[(4*2^(1/3)*x^3)/(9*C[1]^2 - Sqrt[
-256*x^9 + 81*C[1]^4])^(1/3) + (9*C[1]^2 - Sqrt[-256*x^9 + 81*C[1]^4])^(1/3)/2^(
1/3)]]/2}}

Maple raw input

dsolve((x^3-y(x)^4)*diff(y(x),x) = 3*x^2*y(x), y(x))

Maple raw output

[y(x) = RootOf(x^9*_Z^4+3-exp(9/4*_C1)*_Z)*x^3]