4.12.1 $x(2y(x)+x)y^{\prime }(x)+(2x-y(x))y(x)=0$

ODE
$$x(2y(x)+x)y^{\prime }(x)+(2x-y(x))y(x)=0$$ ODE Classification

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

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 0.0315004 (sec), leaf count = 385

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

Maple ✓
cpu = 0.021 (sec), leaf count = 33

$$\{-\frac{5}{3}\ln \left(\frac{3x+y\left(x\right)}{x}\right)-\frac{1}{3}\ln \left(\frac{y\left(x\right)}{x}\right)-\ln \left(x\right)-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ 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),'implicit')

Maple raw output

-5/3*ln((3*x+y(x))/x)-1/3*ln(y(x)/x)-ln(x)-_C1 = 0