##### 4.18.9 $$a x+x y'(x)^2-y(x) y'(x)=0$$

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

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

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

Mathematica
cpu = 1.62749 (sec), leaf count = 167

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

Maple
cpu = 0.186 (sec), leaf count = 50

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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