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 = 0.353271 (sec), leaf count = 59
\[\left \{\left \{y(x)\to -\sqrt {e^{2 c_1}-\left (c_2+x\right ){}^2}-c_2\right \},\left \{y(x)\to \sqrt {e^{2 c_1}-\left (c_2+x\right ){}^2}-c_2\right \}\right \}\]
Maple ✓
cpu = 0.572 (sec), leaf count = 120
\[ \left \{ x-\int ^{y \left ( x \right ) -x}\!{(-{{\rm e}^{2\,{\it \_C1}}}{{\it \_f}}^{2}+1) \left ( {{\rm e}^{2\,{\it \_C1}}}{{\it \_f}}^{2}-{{\rm e}^{{\it \_C1}}}\sqrt {-{{\rm e}^{2\,{\it \_C1}}}{{\it \_f}}^{2}+2}{\it \_f}-2 \right ) ^{-1}}{d{\it \_f}}-{\it \_C2}=0,x-\int ^{y \left ( x \right ) -x}\!{(-{{\rm e}^{2\,{\it \_C1}}}{{\it \_f}}^{2}+1) \left ( {{\rm e}^{2\,{\it \_C1}}}{{\it \_f}}^{2}+{{\rm e}^{{\it \_C1}}}\sqrt {-{{\rm e}^{2\,{\it \_C1}}}{{\it \_f}}^{2}+2}{\it \_f}-2 \right ) ^{-1}}{d{\it \_f}}-{\it \_C2}=0 \right \} \] 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),'implicit')
Maple raw output
x-Intat((-exp(2*_C1)*_f^2+1)/(exp(2*_C1)*_f^2+exp(_C1)*(-exp(2*_C1)*_f^2+2)^(1/2)*_f-2),_f = y(x)-x)-_C2 = 0, x-Intat((-exp(2*_C1)*_f^2+1)/(exp(2*_C1)*_f^2-exp(_C1)*(-exp(2*_C1)*_f^2+2)^(1/2)*_f-2),_f = y(x)-x)-_C2 = 0