2.1727   ODE No. 1727

\[ 2 y(x) y''(x)+y'(x)^2+1=0 \] Mathematica : cpu = 0.335844 (sec), leaf count = 129

DSolve[1 + Derivative[1][y][x]^2 + 2*y[x]*Derivative[2][y][x] == 0,y[x],x]
 

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-e^{2 c_1} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )-\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [e^{2 c_1} \arctan \left (\frac {\sqrt {-\text {$\#$1}+e^{2 c_1}}}{\sqrt {\text {$\#$1}}}\right )+\sqrt {\text {$\#$1}} \sqrt {-\text {$\#$1}+e^{2 c_1}}\& \right ][x+c_2]\right \}\right \}\] Maple : cpu = 0.547 (sec), leaf count = 823

dsolve(2*diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2+1=0,y(x))
 

\[y \left (x \right ) = \frac {\left (-\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} x c_{2}+4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2}-4 c_{1} x \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right ) c_{1}+2 x +2 c_{2}\right ) \tan \left (\operatorname {RootOf}\left (\tan \left (\textit {\_Z} \right )^{2} c_{1}^{2} \textit {\_Z}^{2}-4 \tan \left (\textit {\_Z} \right )^{2} c_{1} c_{2} \textit {\_Z} -4 \tan \left (\textit {\_Z} \right )^{2} c_{1} x \textit {\_Z} +4 \tan \left (\textit {\_Z} \right )^{2} c_{2}^{2}+8 \tan \left (\textit {\_Z} \right )^{2} x c_{2}+4 \tan \left (\textit {\_Z} \right )^{2} x^{2}+c_{1}^{2} \textit {\_Z}^{2}-4 c_{1} \textit {\_Z} c_{2}-4 c_{1} x \textit {\_Z} -c_{1}^{2}+4 c_{2}^{2}+8 c_{2} x +4 x^{2}\right )\right )}{2}+\frac {c_{1}}{2}\]