4.13.4 $$\left (x^2-y(x)^2\right ) y'(x)+x (2 y(x)+x)=0$$

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

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

Book solution method
Exact equation

Mathematica
cpu = 0.48464 (sec), leaf count = 359

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

Maple
cpu = 0.078 (sec), leaf count = 419

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = (1/2*(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3/2)+1)^(1/2))^(1/3)+
2*_C1*x^2/(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3/2)+1)^(1/2))^(1/3))/_C
1^(1/2), y(x) = (-1/4*(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3/2)+1)^(1/2
))^(1/3)-_C1*x^2/(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3/2)+1)^(1/2))^(1
/3)-1/2*I*3^(1/2)*(1/2*(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3/2)+1)^(1/
2))^(1/3)-2*_C1*x^2/(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3/2)+1)^(1/2))
^(1/3)))/_C1^(1/2), y(x) = (-1/4*(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3
/2)+1)^(1/2))^(1/3)-_C1*x^2/(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3/2)+1
)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(
3/2)+1)^(1/2))^(1/3)-2*_C1*x^2/(4+4*x^3*_C1^(3/2)+4*(-3*_C1^3*x^6+2*x^3*_C1^(3/2
)+1)^(1/2))^(1/3)))/_C1^(1/2)]