ODE
\[ x (y(x)+2 x) y'(x)=x^2+x y(x)-y(x)^2 \] ODE Classification
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class B`]]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.046785 (sec), leaf count = 431
\[\left \{\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (\frac {e^{6 c_1}}{x^3}-40 x^3\right )+\text {$\#$1} \left (\frac {2 e^{6 c_1}}{x^2}+10 x^4\right )+\frac {e^{6 c_1}}{x}-x^5\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (\frac {e^{6 c_1}}{x^3}-40 x^3\right )+\text {$\#$1} \left (\frac {2 e^{6 c_1}}{x^2}+10 x^4\right )+\frac {e^{6 c_1}}{x}-x^5\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (\frac {e^{6 c_1}}{x^3}-40 x^3\right )+\text {$\#$1} \left (\frac {2 e^{6 c_1}}{x^2}+10 x^4\right )+\frac {e^{6 c_1}}{x}-x^5\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (\frac {e^{6 c_1}}{x^3}-40 x^3\right )+\text {$\#$1} \left (\frac {2 e^{6 c_1}}{x^2}+10 x^4\right )+\frac {e^{6 c_1}}{x}-x^5\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (\frac {e^{6 c_1}}{x^3}-40 x^3\right )+\text {$\#$1} \left (\frac {2 e^{6 c_1}}{x^2}+10 x^4\right )+\frac {e^{6 c_1}}{x}-x^5\& ,5\right ]\right \}\right \}\]
Maple ✓
cpu = 0.022 (sec), leaf count = 37
\[ \left \{ -{\frac {5}{6}\ln \left ( {\frac {-x+2\,y \left ( x \right ) }{x}} \right ) }+{\frac {1}{3}\ln \left ( {\frac {x+y \left ( x \right ) }{x}} \right ) }-\ln \left ( x \right ) -{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[x*(2*x + y[x])*y'[x] == x^2 + x*y[x] - y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> Root[E^(6*C[1])/x - x^5 + ((2*E^(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^3)*#1^2 + 80*x^2*#1^3 - 80*x*#1^4 + 32*#1^5 & , 1]}, {y[x] -> Root[E^(6*C[1])/x - x^5 + ((2*E^(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^3)*#1^2 + 80*x^2*#1^3 - 80*x*#1^4 + 32*#1^5 & , 2]}, {y[x] -> Root[E^(6*C[1])/x - x^5 + ((2*E^(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^3)*#1^2 + 80*x^2*#1^3 - 80*x*#1^4 + 32*#1^5 & , 3]}, {y[x] -> Root[E^(6*C[1])/x - x^5 + ((2*E^(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^3)*#1^2 + 80*x^2*#1^3 - 80*x*#1^4 + 32*#1^5 & , 4]}, {y[x] -> Root[E^(6*C[1])/x - x^5 + ((2*E^(6*C[1]))/x^2 + 10*x^4)*#1 + (E^(6*C[1])/x^3 - 40*x^3)*#1^2 + 80*x^2*#1^3 - 80*x*#1^4 + 32*#1^5 & , 5]}}
Maple raw input
dsolve(x*(2*x+y(x))*diff(y(x),x) = x^2+x*y(x)-y(x)^2, y(x),'implicit')
Maple raw output
-5/6*ln((-x+2*y(x))/x)+1/3*ln((x+y(x))/x)-ln(x)-_C1 = 0