4.22.15 $x{y}^{\prime }(x{)}^3+(2x-y(x))y^{\prime }(x{)}^2+(2-2y(x))y^{\prime }(x)-y(x)+1=0$

ODE
$$x{y}^{\prime }(x{)}^3+(2x-y(x))y^{\prime }(x{)}^2+(2-2y(x))y^{\prime }(x)-y(x)+1=0$$ ODE Classification

[_dAlembert]

Book solution method
Clairaut’s equation and related types, $f(y-x{y}^{\prime },y^{\prime })=0$

Mathematica ✓
cpu = 526.186 (sec), leaf count = 98

$$\text{Solve}[\{x=c_1{e}^{\frac{1}{2}(\text{K}\mathrm{\$}\text{215689}+1{)}^2+\text{K}\mathrm{\$}\text{215689}}(\text{K}\mathrm{\$}\text{215689}+1{)}^2+\frac{1}{2}(\sqrt{2\pi }(-e^{\frac{1}{2}(\text{K}\mathrm{\$}\text{215689}+2{)}^2})(\text{K}\mathrm{\$}\text{215689}+1{)}^2\text{erf}\left(\frac{\text{K}\mathrm{\$}\text{215689}+2}{\sqrt{2}}\right)-2\text{K}\mathrm{\$}\text{215689}),y(x)=\frac{{\text{K}\mathrm{\$}\text{215689}}^{3}x+2{\text{K}\mathrm{\$}\text{215689}}^{2}x+2\text{K}\mathrm{\$}\text{215689}+1}{(\text{K}\mathrm{\$}\text{215689}+1{)}^2}\},\{y(x),\text{K}\mathrm{\$}\text{215689}\}]$$

Maple ✓
cpu = 0.282 (sec), leaf count = 124

$$\{y\left(x\right)=1,[x\left(\mathit{\_}\mathit{T}\right)={\mathrm{e}}^{\frac{{\mathit{\_}\mathit{T}}^2}{2}}{\left({\mathrm{e}}^{\mathit{\_}\mathit{T}}\right)}^2{(\mathit{\_}\mathit{T}+1)}^2(\int -2\frac{{\mathrm{e}}^{-1/2{\mathit{\_}\mathit{T}}^2}}{{(\mathit{\_}\mathit{T}+1)}^3{\left({\mathrm{e}}^{\mathit{\_}\mathit{T}}\right)}^2}\mathrm{d}\mathit{\_}\mathit{T}+\mathit{\_}\mathit{C}\mathit{1}),y\left(\mathit{\_}\mathit{T}\right)=\frac{1}{{(\mathit{\_}\mathit{T}+1)}^2}({\mathit{\_}\mathit{T}}^2{\left({\mathrm{e}}^{\mathit{\_}\mathit{T}}\right)}^2{\mathrm{e}}^{\frac{{\mathit{\_}\mathit{T}}^2}{2}}(\mathit{\_}\mathit{T}+2){(\mathit{\_}\mathit{T}+1)}^2\int -2\frac{{\mathrm{e}}^{-1/2{\mathit{\_}\mathit{T}}^2}}{{(\mathit{\_}\mathit{T}+1)}^3{\left({\mathrm{e}}^{\mathit{\_}\mathit{T}}\right)}^2}\mathrm{d}\mathit{\_}\mathit{T}+{\mathit{\_}\mathit{T}}^2\mathit{\_}\mathit{C}\mathit{1}{\left({\mathrm{e}}^{\mathit{\_}\mathit{T}}\right)}^2(\mathit{\_}\mathit{T}+2){(\mathit{\_}\mathit{T}+1)}^2{\mathrm{e}}^{\frac{{\mathit{\_}\mathit{T}}^2}{2}}+2\mathit{\_}\mathit{T}+1)]\}$$ Mathematica raw input

DSolve[1 - y[x] + (2 - 2*y[x])*y'[x] + (2*x - y[x])*y'[x]^2 + x*y'[x]^3 == 0,y[x],x]

Mathematica raw output

Solve[{x == E^(K$215689 + (1 + K$215689)^2/2)*(1 + K$215689)^2*C[1] + (-2*K$2156
89 - E^((2 + K$215689)^2/2)*(1 + K$215689)^2*Sqrt[2*Pi]*Erf[(2 + K$215689)/Sqrt[
2]])/2, y[x] == (1 + 2*K$215689 + 2*K$215689^2*x + K$215689^3*x)/(1 + K$215689)^
2}, {y[x], K$215689}]

Maple raw input

dsolve(x*diff(y(x),x)^3+(2*x-y(x))*diff(y(x),x)^2+(2-2*y(x))*diff(y(x),x)+1-y(x) = 0, y(x),'implicit')

Maple raw output

y(x) = 1, [x(_T) = exp(1/2*_T^2)*exp(_T)^2*(_T+1)^2*(Int(-2/(_T+1)^3*exp(-1/2*_T
^2)/exp(_T)^2,_T)+_C1), y(_T) = (_T^2*exp(_T)^2*exp(1/2*_T^2)*(_T+2)*(_T+1)^2*In
t(-2/(_T+1)^3*exp(-1/2*_T^2)/exp(_T)^2,_T)+_T^2*_C1*exp(_T)^2*(_T+2)*(_T+1)^2*ex
p(1/2*_T^2)+2*_T+1)/(_T+1)^2]