##### 4.19.4 $$x^2 y'(x)^2-(2 x y(x)+1) y'(x)+y(x)^2+1=0$$

ODE
$x^2 y'(x)^2-(2 x y(x)+1) y'(x)+y(x)^2+1=0$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

Mathematica
cpu = 1.34883 (sec), leaf count = 101

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

Maple
cpu = 0.043 (sec), leaf count = 42

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

DSolve[1 + y[x]^2 - (1 + 2*x*y[x])*y'[x] + x^2*y'[x]^2 == 0,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x^2*diff(y(x),x)^2-(1+2*x*y(x))*diff(y(x),x)+1+y(x)^2 = 0, y(x))

Maple raw output

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