4.18.7 \(a+x y'(x)^2+y(x) y'(x)=0\)

ODE
\[ a+x y'(x)^2+y(x) y'(x)=0 \] ODE Classification

[[_homogeneous, `class G`], _dAlembert]

Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)

Mathematica
cpu = 25.8856 (sec), leaf count = 1

\[\text {$\$$Aborted}\]

Maple
cpu = 0.02 (sec), leaf count = 39

\[ \left \{ [x \left ({\it \_T} \right ) ={\frac {1}{3\,{{\it \_T}}^{2}} \left (3\,{\it \_C1}\,{{\it \_T}}^{3/2}-a \right ) },y \left ({\it \_T} \right ) =-{\frac {1}{3\,{{\it \_T}}^{2}} \left (3\,{{\it \_T}}^{5/2}{\it \_C1}+2\,{\it \_T}\,a \right ) }] \right \} \] Mathematica raw input

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

Mathematica raw output

$Aborted

Maple raw input

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

Maple raw output

[x(_T) = 1/3*(3*_C1*_T^(3/2)-a)/_T^2, y(_T) = -1/3/_T^2*(3*_T^(5/2)*_C1+2*_T*a)]