4.16.47 \(y'(x)^2-2 x y'(x)+2 y(x)=0\)

ODE
\[ y'(x)^2-2 x y'(x)+2 y(x)=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _Clairaut]

Book solution method
Clairaut’s equation and related types, main form

Mathematica
cpu = 0.306237 (sec), leaf count = 123

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

Maple
cpu = 0.022 (sec), leaf count = 20

\[ \left \{ y \relax (x ) ={\frac {{x}^{2}}{2}},y \relax (x ) =-{\frac {{\it \_C1}\, \left ({\it \_C1}-2\,x \right ) }{2}} \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

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