#### 2.405   ODE No. 405

$a y'(x)^2+y(x) y'(x)-x=0$ Mathematica : cpu = 0.842942 (sec), leaf count = 53

$\text {Solve}\left [\left \{x=\frac {a \text {K\348407} \sin ^{-1}(\text {K\348407})}{\sqrt {1-\text {K\348407}^2}}+\frac {c_1 \text {K\348407}}{\sqrt {1-\text {K\348407}^2}},y(x)=\frac {x}{\text {K\348407}}-a \text {K\348407}\right \},\{y(x),\text {K\348407}\}\right ]$ Maple : cpu = 0.607 (sec), leaf count = 378

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