\[ \boxed { a \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}-y \left ( x \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) -x=0} \]
Mathematica: cpu = 0.864110 (sec), leaf count = 49 \[ \text {Solve}\left [\left \{x=\frac {a \text {K$\$$1211116} \sinh ^{-1}(\text {K$\$$1211116})}{\sqrt {\text {K$\$$1211116}^2+1}}+\frac {c_1 \text {K$\$$1211116}}{\sqrt {\text {K$\$$1211116}^2+1}},y(x)=a \text {K$\$$1211116}-\frac {x}{\text {K$\$$1211116}}\right \},\{y(x),\text {K$\$$1211116}\}\right ] \]
Maple: cpu = 0.577 (sec), leaf count = 266 \[ \left \{ {{\it \_C1} \left ( -y \left ( x \right ) +\sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) {\frac {1}{\sqrt {-{\frac {1 }{{a}^{2}} \left ( 2\,y \left ( x \right ) \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}}-4\,{a}^{2}-4\,ax-2\, \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}+x-{1 \left ( -y \left ( x \right ) + \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) {\it Arcsinh} \left ( {\frac {1}{2\,a} \left ( -y \left ( x \right ) +\sqrt {4 \,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) } \right ) { \frac {1}{\sqrt {-{\frac {1}{{a}^{2}} \left ( 2\,y \left ( x \right ) \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}}-4\,{a}^{2}-4\,a x-2\, \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}=0,{{\it \_C1} \left ( y \left ( x \right ) +\sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) {\frac {1}{\sqrt {{\frac {1}{{a}^{2}} \left ( y \left ( x \right ) \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}}+2\,{a}^{2}+2\,ax+ \left ( y \left ( x \right ) \right ) ^{ 2} \right ) }}}}}+x-{\frac {\sqrt {2}}{2} \left ( y \left ( x \right ) + \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) {\it Arcsinh} \left ( {\frac {1}{2\,a} \left ( y \left ( x \right ) +\sqrt {4\, ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) } \right ) {\frac {1}{\sqrt {{\frac {1}{{a}^{2}} \left ( y \left ( x \right ) \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}}+2\,{a}^{2}+2\,ax+ \left ( y \left ( x \right ) \right ) ^{2} \right ) }}}}}=0 \right \} \]