#### 2.406   ODE No. 406

$a y'(x)^2-y(x) y'(x)-x=0$ Mathematica : cpu = 0.622743 (sec), leaf count = 49

$\text {Solve}\left [\left \{x=\frac {a \text {K\348736} \sinh ^{-1}(\text {K\348736})}{\sqrt {\text {K\348736}^2+1}}+\frac {c_1 \text {K\348736}}{\sqrt {\text {K\348736}^2+1}},y(x)=a \text {K\348736}-\frac {x}{\text {K\348736}}\right \},\{y(x),\text {K\348736}\}\right ]$ Maple : cpu = 0.21 (sec), leaf count = 262

$\left \{ { \left ( -{\frac {\sqrt {2}}{2} \left ( y \left ( x \right ) +\sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) {\it Arcsinh} \left ( {\frac {1}{2\,a} \left ( y \left ( x \right ) +\sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) } \right ) }+x\sqrt {{\frac {1}{{a}^{2}} \left ( y \left ( x \right ) \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}}+2\,{a}^{2}+2\,ax+ \left ( y \left ( x \right ) \right ) ^{2} \right ) }}+{\it \_C1}\,y \left ( x \right ) +{\it \_C1}\,\sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) {\frac {1}{\sqrt {{\frac {1}{{a}^{2}} \left ( y \left ( x \right ) \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}}+ \left ( y \left ( x \right ) \right ) ^{2}+2\,a \left ( x+a \right ) \right ) }}}}}=0,{ \left ( \left ( y \left ( x \right ) -\sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) {\it Arcsinh} \left ( {\frac {1}{2\,a} \left ( -y \left ( x \right ) +\sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) } \right ) +x\sqrt {-2\,{\frac {y \left ( x \right ) \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}}-2\,{a}^{2}-2\,ax- \left ( y \left ( x \right ) \right ) ^{2}}{{a}^{2}}}}-{\it \_C1}\,y \left ( x \right ) +{\it \_C1}\,\sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}} \right ) {\frac {1}{\sqrt {{\frac {1}{{a}^{2}} \left ( -2\,y \left ( x \right ) \sqrt {4\,ax+ \left ( y \left ( x \right ) \right ) ^{2}}+2\, \left ( y \left ( x \right ) \right ) ^{2}+4\,a \left ( x+a \right ) \right ) }}}}}=0 \right \}$