##### 4.14.5 $$x \left (a x y(x)+x^2+2 y(x)^2\right ) y'(x)=y(x)^2 (a x+2 y(x))$$

ODE
$x \left (a x y(x)+x^2+2 y(x)^2\right ) y'(x)=y(x)^2 (a x+2 y(x))$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.467526 (sec), leaf count = 31

$\text {Solve}\left [\frac {a y(x)}{x}+\frac {y(x)^2}{x^2}+\log \left (\frac {y(x)}{x}\right )+\log (x)=c_1,y(x)\right ]$

Maple
cpu = 0.256 (sec), leaf count = 21

$[y \left (x \right ) = {\mathrm e}^{\RootOf \left ({\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+\textit {\_C1} +\textit {\_Z} +\ln \left (x \right )\right )} x]$ Mathematica raw input

DSolve[x*(x^2 + a*x*y[x] + 2*y[x]^2)*y'[x] == y[x]^2*(a*x + 2*y[x]),y[x],x]

Mathematica raw output

Solve[Log[x] + Log[y[x]/x] + (a*y[x])/x + y[x]^2/x^2 == C[1], y[x]]

Maple raw input

dsolve(x*(x^2+a*x*y(x)+2*y(x)^2)*diff(y(x),x) = (a*x+2*y(x))*y(x)^2, y(x))

Maple raw output

[y(x) = exp(RootOf(exp(_Z)^2+a*exp(_Z)+_C1+_Z+ln(x)))*x]