ODE
\[ (x-y(x)) y''(x)=\left (y'(x)+1\right ) \left (y'(x)^2+1\right ) \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 2.17138 (sec), leaf count = 59
\[\left \{\left \{y(x)\to -\sqrt {e^{2 c_1}-(x+c_2){}^2}-c_2\right \},\left \{y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2}-c_2\right \}\right \}\]
Maple ✓
cpu = 2.806 (sec), leaf count = 105
\[\left [y \left (x \right ) = x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}-\frac {\textit {\_C1}^{2} \textit {\_f}^{2}-1}{\textit {\_C1}^{2} \textit {\_f}^{2}+\sqrt {-\textit {\_C1}^{2} \textit {\_f}^{2}+2}\, \textit {\_C1} \textit {\_f} -2}d \textit {\_f} +\textit {\_C2} \right ), y \left (x \right ) = x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {\textit {\_C1}^{2} \textit {\_f}^{2}-1}{-\textit {\_C1}^{2} \textit {\_f}^{2}+\sqrt {-\textit {\_C1}^{2} \textit {\_f}^{2}+2}\, \textit {\_C1} \textit {\_f} +2}d \textit {\_f} +\textit {\_C2} \right )\right ]\] Mathematica raw input
DSolve[(x - y[x])*y''[x] == (1 + y'[x])*(1 + y'[x]^2),y[x],x]
Mathematica raw output
{{y[x] -> -C[2] - Sqrt[E^(2*C[1]) - (x + C[2])^2]}, {y[x] -> -C[2] + Sqrt[E^(2*C[1]) - (x + C[2])^2]}}
Maple raw input
dsolve((x-y(x))*diff(diff(y(x),x),x) = (1+diff(y(x),x))*(1+diff(y(x),x)^2), y(x))
Maple raw output
[y(x) = x+RootOf(-x+Intat(-(_C1^2*_f^2-1)/(_C1^2*_f^2+(-_C1^2*_f^2+2)^(1/2)*_C1*_f-2),_f = _Z)+_C2), y(x) = x+RootOf(-x+Intat((_C1^2*_f^2-1)/(-_C1^2*_f^2+(-_C1^2*_f^2+2)^(1/2)*_C1*_f+2),_f = _Z)+_C2)]