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 = 0.0315004 (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 = 0.021 (sec), leaf count = 33

\[ \left \{ -{\frac {5}{3}\ln \left ({\frac {3\,x+y \relax (x ) }{x}} \right ) }-{\frac {1}{3}\ln \left ({\frac {y \relax (x ) }{x}} \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

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