ODE
\[ x y'(x)^3+(2 x-y(x)) y'(x)^2+(2-2 y(x)) y'(x)-y(x)+1=0 \] ODE Classification
[_dAlembert]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 526.186 (sec), leaf count = 98
\[\text {Solve}\left [\left \{x=c_1 e^{\frac {1}{2} (\text {K$\$$215689}+1)^2+\text {K$\$$215689}} (\text {K$\$$215689}+1)^2+\frac {1}{2} \left (\sqrt {2 \pi } \left (-e^{\frac {1}{2} (\text {K$\$$215689}+2)^2}\right ) (\text {K$\$$215689}+1)^2 \text {erf}\left (\frac {\text {K$\$$215689}+2}{\sqrt {2}}\right )-2 \text {K$\$$215689}\right ),y(x)=\frac {\text {K$\$$215689}^3 x+2 \text {K$\$$215689}^2 x+2 \text {K$\$$215689}+1}{(\text {K$\$$215689}+1)^2}\right \},\{y(x),\text {K$\$$215689}\}\right ]\]
Maple ✓
cpu = 0.282 (sec), leaf count = 124
\[ \left \{ y \left ( x \right ) =1,[x \left ( {\it \_T} \right ) ={{\rm e}^{{\frac {{{\it \_T}}^{2}}{2}}}} \left ( {{\rm e}^{{\it \_T}}} \right ) ^{2} \left ( {\it \_T}+1 \right ) ^{2} \left ( \int \!-2\,{\frac {{{\rm e}^{-1/2\,{{\it \_T}}^{2}}}}{ \left ( {\it \_T}+1 \right ) ^{3} \left ( {{\rm e}^{{\it \_T}}} \right ) ^{2}}}\,{\rm d}{\it \_T}+{\it \_C1} \right ) ,y \left ( {\it \_T} \right ) ={\frac {1}{ \left ( {\it \_T}+1 \right ) ^{2}} \left ( {{\it \_T}}^{2} \left ( {{\rm e}^{{\it \_T}}} \right ) ^{2}{{\rm e}^{{\frac {{{\it \_T}}^{2}}{2}}}} \left ( {\it \_T}+2 \right ) \left ( {\it \_T}+1 \right ) ^{2}\int \!-2\,{\frac {{{\rm e}^{-1/2\,{{\it \_T}}^{2}}}}{ \left ( {\it \_T}+1 \right ) ^{3} \left ( {{\rm e}^{{\it \_T}}} \right ) ^{2}}}\,{\rm d}{\it \_T}+{{\it \_T}}^{2}{\it \_C1}\, \left ( {{\rm e}^{{\it \_T}}} \right ) ^{2} \left ( {\it \_T}+2 \right ) \left ( {\it \_T}+1 \right ) ^{2}{{\rm e}^{{\frac {{{\it \_T}}^{2}}{2}}}}+2\,{\it \_T}+1 \right ) }] \right \} \] 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$215689 - 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*Int(-2/(_T+1)^3*exp(-1/2*_T^2)/exp(_T)^2,_T)+_T^2*_C1*exp(_T)^2*(_T+2)*(_T+1)^2*exp(1/2*_T^2)+2*_T+1)/(_T+1)^2]