4.19.4 $x^2{y}^{\prime }(x{)}^2-(2xy(x)+1)y^{\prime }(x)+y(x{)}^2+1=0$

ODE
$$x^2{y}^{\prime }(x{)}^2-(2xy(x)+1)y^{\prime }(x)+y(x{)}^2+1=0$$ ODE Classification

[[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

Book solution method
Clairaut’s equation and related types, $f(y-x{y}^{\prime },y^{\prime })=0$

Mathematica ✓
cpu = 0.498578 (sec), leaf count = 101

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

Maple ✓
cpu = 0.045 (sec), leaf count = 42

$$\{y\left(x\right)=\frac{4x^2-1}{4x},y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}x-\sqrt{\mathit{\_}\mathit{C}\mathit{1}-1},y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}x+\sqrt{\mathit{\_}\mathit{C}\mathit{1}-1}\}$$ 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*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),'implicit')

Maple raw output

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