##### 4.13.15 $$\left (x^3-y(x)^2+2 y(x)\right ) y'(x)+3 x^2 y(x)=0$$

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

[_exact, _rational, [_1st_order, _with_symmetry_[F(x)*G(y),0]]]

Book solution method
Exact equation

Mathematica
cpu = 0.360865 (sec), leaf count = 404

$\left \{\left \{y(x)\to -\frac {\sqrt [3]{2} \left (x^3+1\right )}{\sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}-\frac {\sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{\sqrt [3]{2}}+1\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (x^3+1\right )}{2^{2/3} \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{2 \sqrt [3]{2}}+1\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (x^3+1\right )}{2^{2/3} \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-3 x^3+\sqrt {-4 x^9-3 x^6-18 c_1 x^3+3 c_1 (-4+3 c_1)}-2+3 c_1}}{2 \sqrt [3]{2}}+1\right \}\right \}$

Maple
cpu = 0.028 (sec), leaf count = 493

$\left [y \left (x \right ) = \frac {\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}{2}-\frac {2 \left (-x^{3}-1\right )}{\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}+1, y \left (x \right ) = -\frac {\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}{4}+\frac {-x^{3}-1}{\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}+1-\frac {i \sqrt {3}\, \left (\frac {\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}{2}+\frac {-2 x^{3}-2}{\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}{4}+\frac {-x^{3}-1}{\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (\frac {\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}{2}+\frac {-2 x^{3}-2}{\left (12 x^{3}+12 \textit {\_C1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}+12 \textit {\_C1}}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> 1 - (2^(1/3)*(1 + x^3))/(-2 - 3*x^3 + 3*C[1] + Sqrt[-3*x^6 - 4*x^9 - 1
8*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3) - (-2 - 3*x^3 + 3*C[1] + Sqrt[-3*x^6 -
 4*x^9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3)/2^(1/3)}, {y[x] -> 1 + ((1 +
 I*Sqrt[3])*(1 + x^3))/(2^(2/3)*(-2 - 3*x^3 + 3*C[1] + Sqrt[-3*x^6 - 4*x^9 - 18*
x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3)) + ((1 - I*Sqrt[3])*(-2 - 3*x^3 + 3*C[1]
 + Sqrt[-3*x^6 - 4*x^9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3))/(2*2^(1/3))
}, {y[x] -> 1 + ((1 - I*Sqrt[3])*(1 + x^3))/(2^(2/3)*(-2 - 3*x^3 + 3*C[1] + Sqrt
[-3*x^6 - 4*x^9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])])^(1/3)) + ((1 + I*Sqrt[3])
*(-2 - 3*x^3 + 3*C[1] + Sqrt[-3*x^6 - 4*x^9 - 18*x^3*C[1] + 3*C[1]*(-4 + 3*C[1])
])^(1/3))/(2*2^(1/3))}}

Maple raw input

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

Maple raw output

[y(x) = 1/2*(12*x^3+12*_C1+8+4*(-4*x^9-3*x^6+18*_C1*x^3+9*_C1^2+12*_C1)^(1/2))^(
1/3)-2*(-x^3-1)/(12*x^3+12*_C1+8+4*(-4*x^9-3*x^6+18*_C1*x^3+9*_C1^2+12*_C1)^(1/2
))^(1/3)+1, y(x) = -1/4*(12*x^3+12*_C1+8+4*(-4*x^9-3*x^6+18*_C1*x^3+9*_C1^2+12*_
C1)^(1/2))^(1/3)+(-x^3-1)/(12*x^3+12*_C1+8+4*(-4*x^9-3*x^6+18*_C1*x^3+9*_C1^2+12
*_C1)^(1/2))^(1/3)+1-1/2*I*3^(1/2)*(1/2*(12*x^3+12*_C1+8+4*(-4*x^9-3*x^6+18*_C1*
x^3+9*_C1^2+12*_C1)^(1/2))^(1/3)+2*(-x^3-1)/(12*x^3+12*_C1+8+4*(-4*x^9-3*x^6+18*
_C1*x^3+9*_C1^2+12*_C1)^(1/2))^(1/3)), y(x) = -1/4*(12*x^3+12*_C1+8+4*(-4*x^9-3*
x^6+18*_C1*x^3+9*_C1^2+12*_C1)^(1/2))^(1/3)+(-x^3-1)/(12*x^3+12*_C1+8+4*(-4*x^9-
3*x^6+18*_C1*x^3+9*_C1^2+12*_C1)^(1/2))^(1/3)+1+1/2*I*3^(1/2)*(1/2*(12*x^3+12*_C
1+8+4*(-4*x^9-3*x^6+18*_C1*x^3+9*_C1^2+12*_C1)^(1/2))^(1/3)+2*(-x^3-1)/(12*x^3+1
2*_C1+8+4*(-4*x^9-3*x^6+18*_C1*x^3+9*_C1^2+12*_C1)^(1/2))^(1/3))]