##### 4.18.16 $$(a-y(x)+x) y'(x)+x y'(x)^2-y(x)=0$$

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

[[_1st_order, _with_linear_symmetries], _rational, _dAlembert]

Book solution method
Clairaut’s equation and related types, $$f(y-x y', y')=0$$

Mathematica
cpu = 0.155824 (sec), leaf count = 18

$\left \{\left \{y(x)\to c_1 \left (x+\frac {a}{1+c_1}\right )\right \}\right \}$

Maple
cpu = 0.049 (sec), leaf count = 63

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

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

Mathematica raw output

{{y[x] -> C[1]*(x + a/(1 + C[1]))}}

Maple raw input

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

Maple raw output

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