\[ (y(x)-2 x) y'(x)^2-2 (x-1) y'(x)+y(x)-2=0 \] ✓ Mathematica : cpu = 0.307533 (sec), leaf count = 165
DSolve[-2 + y[x] - 2*(-1 + x)*Derivative[1][y][x] + (-2*x + y[x])*Derivative[1][y][x]^2 == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {1}{2} \left (-\sqrt {-4 e^{c_1} x+4 e^{c_1}-e^{2 c_1}}+4-e^{c_1}\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\sqrt {-4 e^{c_1} x+4 e^{c_1}-e^{2 c_1}}+4-e^{c_1}\right )\right \},\left \{y(x)\to -\sqrt {-2 e^{c_1} x+2 e^{c_1}-e^{2 c_1}}+2-e^{c_1}\right \},\left \{y(x)\to \sqrt {-2 e^{c_1} x+2 e^{c_1}-e^{2 c_1}}+2-e^{c_1}\right \}\right \}\] ✓ Maple : cpu = 0.706 (sec), leaf count = 71
dsolve((y(x)-2*x)*diff(y(x),x)^2-2*(x-1)*diff(y(x),x)+y(x)-2 = 0,y(x))
\[y \left (x \right ) = -\sqrt {2}\, x +\sqrt {2}+x +1\]