\[ y''(x) \left (a \sqrt {y'(x)^2+1}-x y'(x)\right )-y'(x)^2-1=0 \] ✓ Mathematica : cpu = 0.638173 (sec), leaf count = 347
DSolve[-1 - Derivative[1][y][x]^2 + (-(x*Derivative[1][y][x]) + a*Sqrt[1 + Derivative[1][y][x]^2])*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {-2 \sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x \log \left (-c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 (-x)+a x^2\right )+c_1 x \log \left (c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 x+a x^2\right )+c_1 x \log (x-a)-c_1 x \log (x (x-a))-c_1 x \log (a+x)-c_1 x \log (x (a+x))}{2 x}+c_2\right \},\left \{y(x)\to c_2-\frac {-2 \sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x \log \left (-c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 (-x)+a x^2\right )+c_1 x \log \left (c_1 \left (\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )}+c_1 x\right )+a^2 x+a x^2\right )+c_1 (-x) \log (x-a)-c_1 x \log (x (x-a))+c_1 x \log (a+x)-c_1 x \log (x (a+x))}{2 x}\right \}\right \}\] ✓ Maple : cpu = 1.095 (sec), leaf count = 96
dsolve((a*(diff(y(x),x)^2+1)^(1/2)-x*diff(y(x),x))*diff(diff(y(x),x),x)-diff(y(x),x)^2-1=0,y(x))
\[y \left (x \right ) = \int \frac {-c_{1} a^{2}+x \sqrt {a^{2} \left (a^{2}-x^{2}+c_{1}^{2}\right )}}{a^{3}-a \,x^{2}}d x +c_{2}\]