##### 4.16.27 $$a (x-y(x))+y'(x)^2-2 y'(x)=0$$

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

[[_homogeneous, class C], _dAlembert]

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

Mathematica
cpu = 0.392074 (sec), leaf count = 73

$\left \{\left \{y(x)\to \frac {1}{4} a \left (x^2-2 \sqrt {2} c_1 x+2 c_1{}^2\right )-\frac {1}{a}+x\right \},\left \{y(x)\to \frac {1}{4} a \left (x^2+2 \sqrt {2} c_1 x+2 c_1{}^2\right )-\frac {1}{a}+x\right \}\right \}$

Maple
cpu = 0.035 (sec), leaf count = 42

$\left [y \left (x \right ) = \frac {a x -1}{a}, y \left (x \right ) = x +\frac {\frac {\left (-x +\textit {\_C1} \right )^{2} a^{2}}{4}+\left (-x +\textit {\_C1} \right ) a}{a}\right ]$ Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

[y(x) = (a*x-1)/a, y(x) = x+(1/4*(-x+_C1)^2*a^2+(-x+_C1)*a)/a]