\[ 2 (y(x)-a) y''(x)+y'(x)^2+1=0 \] ✓ Mathematica : cpu = 0.688712 (sec), leaf count = 251
DSolve[1 + Derivative[1][y][x]^2 + 2*(-a + y[x])*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \sin ^{-1}\left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \sin ^{-1}\left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 1.405 (sec), leaf count = 117
dsolve(2*(y(x)-a)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1=0,y(x))
\[\frac {\arctan \left (\frac {y \left (x \right )-a -\frac {c_{1}}{2}}{\sqrt {-\left (-y \left (x \right )+a \right ) \left (a +c_{1}-y \left (x \right )\right )}}\right ) c_{1}}{2}-x -c_{2}-\sqrt {-\left (-y \left (x \right )+a \right ) \left (a +c_{1}-y \left (x \right )\right )} = 0\]