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 = 0.498578 (sec), leaf count = 101

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

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

\[ \left \{ y \relax (x ) ={\frac {4\,{x}^{2}-1}{4\,x}},y \relax (x ) ={\it \_C1}\,x-\sqrt {{\it \_C1}-1},y \relax (x ) ={\it \_C1}\,x+\sqrt {{\it \_C1}-1} \right \} \] 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*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)