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 = 0.309555 (sec), leaf count = 431
\[\left \{\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [32 \text {$\#$1}^5-80 \text {$\#$1}^4 x+80 \text {$\#$1}^3 x^2+\text {$\#$1}^2 \left (-40 x^3+\frac {e^{6 c_1}}{x^3}\right )+\text {$\#$1} \left (10 x^4+\frac {2 e^{6 c_1}}{x^2}\right )-x^5+\frac {e^{6 c_1}}{x}\& ,5\right ]\right \}\right \}\]
Maple ✓
cpu = 0.291 (sec), leaf count = 63
\[\left [y \left (x \right ) = -\frac {x \left (-\RootOf \left (3 \textit {\_Z}^{15}+\textit {\_Z}^{9}-2 \textit {\_C1} \,x^{3}\right )^{9}-\textit {\_C1} \,x^{3}\right )}{-\RootOf \left (3 \textit {\_Z}^{15}+\textit {\_Z}^{9}-2 \textit {\_C1} \,x^{3}\right )^{9}+2 \textit {\_C1} \,x^{3}}\right ]\] 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))
Maple raw output
[y(x) = -x*(-RootOf(3*_Z^15+_Z^9-2*_C1*x^3)^9-_C1*x^3)/(-RootOf(3*_Z^15+_Z^9-2*_C1*x^3)^9+2*_C1*x^3)]