##### 4.18.12 $$a y(x)+x y'(x)^2-y(x) y'(x)=0$$

ODE
$a y(x)+x y'(x)^2-y(x) y'(x)=0$ ODE Classiﬁcation

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

Book solution method
No Missing Variables ODE, Solve for $$x$$

Mathematica
cpu = 1.18733 (sec), leaf count = 215

$\left \{\text {Solve}\left [-\frac {\sqrt {\frac {y(x)}{x}-4 a} \left (\sqrt {\frac {y(x)}{x}}-\frac {4 a^{3/2} x \sqrt {4-\frac {y(x)}{a x}} \sin ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}}}{2 \sqrt {a}}\right )}{y(x)-4 a x}\right )+\frac {y(x)}{x}}{4 a}=\frac {\log (x)}{2}+c_1,y(x)\right ],\text {Solve}\left [\frac {\sqrt {\frac {y(x)}{x}-4 a} \left (\frac {4 a^{3/2} x \sqrt {4-\frac {y(x)}{a x}} \sin ^{-1}\left (\frac {\sqrt {\frac {y(x)}{x}}}{2 \sqrt {a}}\right )}{y(x)-4 a x}-\sqrt {\frac {y(x)}{x}}\right )+\frac {y(x)}{x}}{4 a}+\frac {\log (x)}{2}=c_1,y(x)\right ]\right \}$

Maple
cpu = 0.17 (sec), leaf count = 55

$\left [y \left (x \right ) = 0, y \left (x \right ) = -\frac {\left (-\LambertW \left (-\frac {x \,{\mathrm e}}{\textit {\_C1} a}\right )+1\right )^{2} a^{2} x}{-\left (-\LambertW \left (-\frac {x \,{\mathrm e}}{\textit {\_C1} a}\right )+1\right ) a +a}\right ]$ Mathematica raw input

DSolve[a*y[x] - y[x]*y'[x] + x*y'[x]^2 == 0,y[x],x]

Mathematica raw output

{Solve[-1/4*(y[x]/x + Sqrt[-4*a + y[x]/x]*(Sqrt[y[x]/x] - (4*a^(3/2)*x*ArcSin[Sq
rt[y[x]/x]/(2*Sqrt[a])]*Sqrt[4 - y[x]/(a*x)])/(-4*a*x + y[x])))/a == C[1] + Log[
x]/2, y[x]], Solve[Log[x]/2 + (y[x]/x + Sqrt[-4*a + y[x]/x]*(-Sqrt[y[x]/x] + (4*
a^(3/2)*x*ArcSin[Sqrt[y[x]/x]/(2*Sqrt[a])]*Sqrt[4 - y[x]/(a*x)])/(-4*a*x + y[x])
))/(4*a) == C[1], y[x]]}

Maple raw input

dsolve(x*diff(y(x),x)^2-y(x)*diff(y(x),x)+a*y(x) = 0, y(x))

Maple raw output

[y(x) = 0, y(x) = -(-LambertW(-x/_C1/a*exp(1))+1)^2*a^2*x/(-(-LambertW(-x/_C1/a*
exp(1))+1)*a+a)]