##### 4.12.1 $$x (2 y(x)+x) y'(x)+(2 x-y(x)) y(x)=0$$

ODE
$x (2 y(x)+x) y'(x)+(2 x-y(x)) y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, [_Abel, 2nd type, class B]]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.289674 (sec), leaf count = 385

$\left \{\left \{y(x)\to \text {Root}\left [\text {\#1}^6+15 \text {\#1}^5 x+90 \text {\#1}^4 x^2+270 \text {\#1}^3 x^3+405 \text {\#1}^2 x^4+243 \text {\#1} x^5-e^{3 c_1} x^3\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {\#1}^6+15 \text {\#1}^5 x+90 \text {\#1}^4 x^2+270 \text {\#1}^3 x^3+405 \text {\#1}^2 x^4+243 \text {\#1} x^5-e^{3 c_1} x^3\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {\#1}^6+15 \text {\#1}^5 x+90 \text {\#1}^4 x^2+270 \text {\#1}^3 x^3+405 \text {\#1}^2 x^4+243 \text {\#1} x^5-e^{3 c_1} x^3\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {\#1}^6+15 \text {\#1}^5 x+90 \text {\#1}^4 x^2+270 \text {\#1}^3 x^3+405 \text {\#1}^2 x^4+243 \text {\#1} x^5-e^{3 c_1} x^3\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {\#1}^6+15 \text {\#1}^5 x+90 \text {\#1}^4 x^2+270 \text {\#1}^3 x^3+405 \text {\#1}^2 x^4+243 \text {\#1} x^5-e^{3 c_1} x^3\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [\text {\#1}^6+15 \text {\#1}^5 x+90 \text {\#1}^4 x^2+270 \text {\#1}^3 x^3+405 \text {\#1}^2 x^4+243 \text {\#1} x^5-e^{3 c_1} x^3\& ,6\right ]\right \}\right \}$

Maple
cpu = 0.256 (sec), leaf count = 33

$\left [y \left (x \right ) = \frac {\RootOf \left (\textit {\_Z}^{18}+3 x^{3} \textit {\_C1} \,\textit {\_Z}^{3}-\textit {\_C1} \,x^{3}\right )^{15}}{\textit {\_C1} \,x^{2}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> Root[-(E^(3*C[1])*x^3) + 243*x^5*#1 + 405*x^4*#1^2 + 270*x^3*#1^3 + 90
*x^2*#1^4 + 15*x*#1^5 + #1^6 & , 1]}, {y[x] -> Root[-(E^(3*C[1])*x^3) + 243*x^5*
#1 + 405*x^4*#1^2 + 270*x^3*#1^3 + 90*x^2*#1^4 + 15*x*#1^5 + #1^6 & , 2]}, {y[x]
 -> Root[-(E^(3*C[1])*x^3) + 243*x^5*#1 + 405*x^4*#1^2 + 270*x^3*#1^3 + 90*x^2*#
1^4 + 15*x*#1^5 + #1^6 & , 3]}, {y[x] -> Root[-(E^(3*C[1])*x^3) + 243*x^5*#1 + 4
05*x^4*#1^2 + 270*x^3*#1^3 + 90*x^2*#1^4 + 15*x*#1^5 + #1^6 & , 4]}, {y[x] -> Ro
ot[-(E^(3*C[1])*x^3) + 243*x^5*#1 + 405*x^4*#1^2 + 270*x^3*#1^3 + 90*x^2*#1^4 +
15*x*#1^5 + #1^6 & , 5]}, {y[x] -> Root[-(E^(3*C[1])*x^3) + 243*x^5*#1 + 405*x^4
*#1^2 + 270*x^3*#1^3 + 90*x^2*#1^4 + 15*x*#1^5 + #1^6 & , 6]}}

Maple raw input

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

Maple raw output

[y(x) = 1/_C1*RootOf(_Z^18+3*_C1*_Z^3*x^3-_C1*x^3)^15/x^2]