ODE
\[ \left (y(x)^2+1\right ) y''(x)+y'(x) \left (y'(x)^2+1\right )=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.4809 (sec), leaf count = 42
\[\left \{\left \{y(x)\to \csc (c_1) \sec (c_1) W\left (\sin (c_1) e^{-\left ((x+c_2) \cos ^2(c_1)\right )-\sin ^2(c_1)}\right )+\tan (c_1)\right \}\right \}\]
Maple ✓
cpu = 3.087 (sec), leaf count = 116
\[\left [y \left (x \right ) = \textit {\_C1}, y \left (x \right ) = \frac {i \textit {\_C1} -i-{\mathrm e}^{-\frac {-\textit {\_C1}^{2}+2 \textit {\_C1} +\textit {\_C1}^{2} \textit {\_C2} +\textit {\_C1}^{2} x -1+4 \LambertW \left (-\frac {i {\mathrm e}^{-\frac {\textit {\_C2} \textit {\_C1}}{4}} {\mathrm e}^{-\frac {\textit {\_C1} x}{4}} {\mathrm e}^{\frac {\textit {\_C1}}{4}} {\mathrm e}^{-\frac {\textit {\_C2}}{2}} {\mathrm e}^{-\frac {x}{2}} {\mathrm e}^{-\frac {1}{2}} {\mathrm e}^{-\frac {\textit {\_C2}}{4 \textit {\_C1}}} {\mathrm e}^{-\frac {x}{4 \textit {\_C1}}} {\mathrm e}^{\frac {1}{4 \textit {\_C1}}} \left (\textit {\_C1} -1\right )}{4 \textit {\_C1}}\right ) \textit {\_C1} +2 \textit {\_C2} \textit {\_C1} +2 \textit {\_C1} x +\textit {\_C2} +x}{4 \textit {\_C1}}}}{\textit {\_C1} +1}\right ]\] Mathematica raw input
DSolve[y'[x]*(1 + y'[x]^2) + (1 + y[x]^2)*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> Csc[C[1]]*ProductLog[E^(-((x + C[2])*Cos[C[1]]^2) - Sin[C[1]]^2)*Sin[C[1]]]*Sec[C[1]] + Tan[C[1]]}}
Maple raw input
dsolve((1+y(x)^2)*diff(diff(y(x),x),x)+diff(y(x),x)*(1+diff(y(x),x)^2) = 0, y(x))
Maple raw output
[y(x) = _C1, y(x) = (I*_C1-I-exp(-1/4*(-_C1^2+2*_C1+_C1^2*_C2+_C1^2*x-1+4*LambertW(-1/4*I*exp(-1/4*_C2*_C1)*exp(-1/4*_C1*x)*exp(1/4*_C1)*exp(-1/2*_C2)*exp(-1/2*x)*exp(-1/2)*exp(-1/4/_C1*_C2)*exp(-1/4*x/_C1)*exp(1/4/_C1)*(_C1-1)/_C1)*_C1+2*_C2*_C1+2*_C1*x+_C2+x)/_C1))/(_C1+1)]