2.1727 ODE No. 1727
\[ 2 y(x) y''(x)+y'(x)^2+1=0 \]
✓ Mathematica : cpu = 1.43328 (sec), leaf count = 482
DSolve[1 + Derivative[1][y][x]^2 + 2*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [2 e^{2 c_1} \tan ^{-1}\left (\frac {1-\sqrt {\text {$\#$1}}}{\sqrt {-1+e^{2 c_1}}-\sqrt {-\text {$\#$1}+e^{2 c_1}}}\right )-\frac {\left (\sqrt {\text {$\#$1}}-1\right ) \left (\sqrt {-1+e^{2 c_1}}-\sqrt {-\text {$\#$1}+e^{2 c_1}}\right ) \left (e^{2 c_1} \left (\text {$\#$1}+\sqrt {\text {$\#$1}}+\sqrt {-1+e^{2 c_1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}+1\right )-2 \left (\text {$\#$1}+\sqrt {\text {$\#$1}} \sqrt {-1+e^{2 c_1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\right )-e^{4 c_1}\right )}{\left (\sqrt {\text {$\#$1}}+\sqrt {-1+e^{2 c_1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}-e^{2 c_1}\right ){}^2}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\left (\sqrt {\text {$\#$1}}-1\right ) \left (\sqrt {-1+e^{2 c_1}}-\sqrt {-\text {$\#$1}+e^{2 c_1}}\right ) \left (e^{2 c_1} \left (\text {$\#$1}+\sqrt {\text {$\#$1}}+\sqrt {-1+e^{2 c_1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}+1\right )-2 \left (\text {$\#$1}+\sqrt {\text {$\#$1}} \sqrt {-1+e^{2 c_1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\right )-e^{4 c_1}\right )}{\left (\sqrt {\text {$\#$1}}+\sqrt {-1+e^{2 c_1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}-e^{2 c_1}\right ){}^2}-2 e^{2 c_1} \tan ^{-1}\left (\frac {1-\sqrt {\text {$\#$1}}}{\sqrt {-1+e^{2 c_1}}-\sqrt {-\text {$\#$1}+e^{2 c_1}}}\right )\& \right ][x+c_2]\right \}\right \}\]
✓ Maple : cpu = 0.538 (sec), leaf count = 823
dsolve(2*diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2+1=0,y(x))
\[y \left (x \right ) = \frac {\left (-\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} x c_{2} +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right ) c_{1} +2 x +2 c_{2} \right ) \tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} x c_{2} +4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2} -4 c_{1} \textit {\_Z} x -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right )}{2}+\frac {c_{1}}{2}\]