4.18.22 $x{y}^{\prime }(x{)}^2-2y(x)y^{\prime }(x)+2y(x)+x=0$

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

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

Book solution method
Change of variable

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

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

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

$$\{{\left(y\left(x\right)\right)}^2-2xy\left(x\right)-x^2=0,[x\left(\mathit{\_}\mathit{T}\right)=(\mathit{\_}\mathit{T}-1)\mathit{\_}\mathit{C}\mathit{1},y\left(\mathit{\_}\mathit{T}\right)=\frac{({\mathit{\_}\mathit{T}}^2+1)(\mathit{\_}\mathit{T}-1)\mathit{\_}\mathit{C}\mathit{1}}{2\mathit{\_}\mathit{T}-2}]\}$$ 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)]