##### 4.19.5 $$-(a+2 x y(x)) y'(x)+x^2 y'(x)^2+y(x)^2=0$$

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

[[_homogeneous, class G], _rational, _Clairaut]

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

Mathematica
cpu = 0.555914 (sec), leaf count = 47

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

Maple
cpu = 0.115 (sec), leaf count = 36

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

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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