#### 2.472   ODE No. 472

$(y(x)+x) y'(x)^2+2 x y'(x)-y(x)=0$ Mathematica : cpu = 0.39306 (sec), leaf count = 269

$\left \{\left \{y(x)\to -\frac {2 \sqrt {-\sqrt {3} x \sinh (c_1)-\sqrt {3} x \cosh (c_1)+\sinh (2 c_1)+\cosh (2 c_1)}}{\sqrt {3}}-\frac {\sinh (c_1)}{\sqrt {3}}-\frac {\cosh (c_1)}{\sqrt {3}}\right \},\left \{y(x)\to \frac {2 \sqrt {-\sqrt {3} x \sinh (c_1)-\sqrt {3} x \cosh (c_1)+\sinh (2 c_1)+\cosh (2 c_1)}}{\sqrt {3}}-\frac {\sinh (c_1)}{\sqrt {3}}-\frac {\cosh (c_1)}{\sqrt {3}}\right \},\left \{y(x)\to -\frac {2 \sqrt {\sqrt {3} x \sinh (c_1)+\sqrt {3} x \cosh (c_1)+\sinh (2 c_1)+\cosh (2 c_1)}}{\sqrt {3}}+\frac {\sinh (c_1)}{\sqrt {3}}+\frac {\cosh (c_1)}{\sqrt {3}}\right \},\left \{y(x)\to \frac {2 \sqrt {\sqrt {3} x \sinh (c_1)+\sqrt {3} x \cosh (c_1)+\sinh (2 c_1)+\cosh (2 c_1)}}{\sqrt {3}}+\frac {\sinh (c_1)}{\sqrt {3}}+\frac {\cosh (c_1)}{\sqrt {3}}\right \}\right \}$ Maple : cpu = 1.371 (sec), leaf count = 121

$\left \{ \ln \left ( x \right ) -{\it Artanh} \left ( {\frac {y \left ( x \right ) +2\,x}{2\,x}{\frac {1}{\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}+xy \left ( x \right ) +{x}^{2}}{{x}^{2}}}}}}} \right ) +\ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) -{\it \_C1}=0,\ln \left ( x \right ) +{\it Artanh} \left ( {\frac {y \left ( x \right ) +2\,x}{2\,x}{\frac {1}{\sqrt {{\frac { \left ( y \left ( x \right ) \right ) ^{2}+xy \left ( x \right ) +{x}^{2}}{{x}^{2}}}}}}} \right ) +\ln \left ( {\frac {y \left ( x \right ) }{x}} \right ) -{\it \_C1}=0,y \left ( x \right ) =-{\frac { \left ( 1+i\sqrt {3} \right ) x}{2}},y \left ( x \right ) ={\frac { \left ( i\sqrt {3}-1 \right ) x}{2}} \right \}$