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

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

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
Change of variable

Mathematica
cpu = 0.0752985 (sec), leaf count = 51

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

Maple
cpu = 0.032 (sec), leaf count = 47

\[ \left \{ \left (y \relax (x ) \right ) ^{2}-2\,xy \relax (x ) -{x}^{2}=0,[x \left ({\it \_T} \right ) = \left ({\it \_T}-1 \right ) {\it \_C1},y \left ({\it \_T} \right ) ={\frac { \left ({{\it \_T}}^{2}+1 \right ) \left ({\it \_T}-1 \right ) {\it \_C1}}{2\,{\it \_T}-2}}] \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

y(x)^2-2*x*y(x)-x^2 = 0, [x(_T) = (_T-1)*_C1, y(_T) = (_T^2+1)*(_T-1)*_C1/(2*_T-
2)]