ODE
\[ \left (y(x) y''(x)+y'(x)^2+1\right )^2=\left (y'(x)^2+1\right )^3 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 9.80387 (sec), leaf count = 121
\[\left \{\left \{y(x)\to -\sqrt {e^{2 c_1}-(x+c_2){}^2}-e^{c_1}\right \},\left \{y(x)\to e^{c_1}-\sqrt {e^{2 c_1}-(x+c_2){}^2}\right \},\left \{y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2}-e^{c_1}\right \},\left \{y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2}+e^{c_1}\right \}\right \}\]
Maple ✓
cpu = 9.384 (sec), leaf count = 131
\[\left [y \left (x \right ) = -i x +\textit {\_C1}, y \left (x \right ) = i x +\textit {\_C1}, y \left (x \right ) = 0, y \left (x \right ) = -\textit {\_C1} -\sqrt {\textit {\_C1}^{2}-\textit {\_C2}^{2}-2 \textit {\_C2} x -x^{2}}, y \left (x \right ) = -\textit {\_C1} +\sqrt {\textit {\_C1}^{2}-\textit {\_C2}^{2}-2 \textit {\_C2} x -x^{2}}, y \left (x \right ) = \textit {\_C1} -\sqrt {\textit {\_C1}^{2}-\textit {\_C2}^{2}-2 \textit {\_C2} x -x^{2}}, y \left (x \right ) = \textit {\_C1} +\sqrt {\textit {\_C1}^{2}-\textit {\_C2}^{2}-2 \textit {\_C2} x -x^{2}}\right ]\] Mathematica raw input
DSolve[(1 + y'[x]^2 + y[x]*y''[x])^2 == (1 + y'[x]^2)^3,y[x],x]
Mathematica raw output
{{y[x] -> -E^C[1] - Sqrt[E^(2*C[1]) - (x + C[2])^2]}, {y[x] -> E^C[1] - Sqrt[E^(2*C[1]) - (x + C[2])^2]}, {y[x] -> -E^C[1] + Sqrt[E^(2*C[1]) - (x + C[2])^2]}, {y[x] -> E^C[1] + Sqrt[E^(2*C[1]) - (x + C[2])^2]}}
Maple raw input
dsolve((1+diff(y(x),x)^2+y(x)*diff(diff(y(x),x),x))^2 = (1+diff(y(x),x)^2)^3, y(x))
Maple raw output
[y(x) = -I*x+_C1, y(x) = I*x+_C1, y(x) = 0, y(x) = -_C1-(_C1^2-_C2^2-2*_C2*x-x^2)^(1/2), y(x) = -_C1+(_C1^2-_C2^2-2*_C2*x-x^2)^(1/2), y(x) = _C1-(_C1^2-_C2^2-2*_C2*x-x^2)^(1/2), y(x) = _C1+(_C1^2-_C2^2-2*_C2*x-x^2)^(1/2)]