4.15.10 $(x^3-y(x{)}^4)y^{\prime }(x)=3x^{2}y(x)$

ODE
$$(x^3-y(x{)}^4)y^{\prime }(x)=3x^{2}y(x)$$ ODE Classification

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

Book solution method
Exact equation, integrating factor

Mathematica ✓
cpu = 0.0724627 (sec), leaf count = 1021

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

Maple ✓
cpu = 0.019 (sec), leaf count = 35

$$\{\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}+\frac{4}{9}\ln \left(\frac{{\left(y\left(x\right)\right)}^4+3x^3}{x^3}\right)-\frac{4}{9}\ln \left(y\left(x\right)x^{-\frac{3}{4}}\right)=0\}$$ 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] -> -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 + 8
1*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}}

Maple raw input

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

Maple raw output

ln(x)-_C1+4/9*ln((y(x)^4+3*x^3)/x^3)-4/9*ln(y(x)/x^(3/4)) = 0