2.493   ODE No. 493

$\left (a^2-2 a x+y(x)^2\right ) y'(x)^2+2 a y(x) y'(x)+y(x)^2=0$ Mathematica : cpu = 7.08136 (sec), leaf count = 553

$\left \{\text {Solve}\left [\left \{y(x)=\frac {-\sqrt {-a \text {K\4089}^2 \left (a \text {K\4089}^2-2 \text {K\4089}^2 x-2 x\right )}-a \text {K\4089}}{\text {K\4089}^2+1},x=\frac {a \text {K\4089}^2+a \text {K\4089}^2 \log ^2(\text {K\4089})+a \text {K\4089}^2 \log ^2\left (\sqrt {\text {K\4089}^2+1}+1\right )+a \log ^2\left (\sqrt {\text {K\4089}^2+1}+1\right )-2 a \text {K\4089}^2 \log (\text {K\4089}) \log \left (\sqrt {\text {K\4089}^2+1}+1\right )+2 a \sqrt {\text {K\4089}^2+1} \log (\text {K\4089})-2 a \log (\text {K\4089}) \log \left (\sqrt {\text {K\4089}^2+1}+1\right )-2 a \sqrt {\text {K\4089}^2+1} \log \left (\sqrt {\text {K\4089}^2+1}+1\right )+a c_1{}^2 \text {K\4089}^2-2 a c_1 \sqrt {\text {K\4089}^2+1}-2 a c_1 \text {K\4089}^2 \log (\text {K\4089})+2 a c_1 \text {K\4089}^2 \log \left (\sqrt {\text {K\4089}^2+1}+1\right )+2 a c_1 \log \left (\sqrt {\text {K\4089}^2+1}+1\right )+a \log ^2(\text {K\4089})-2 a c_1 \log (\text {K\4089})+a+a c_1{}^2}{2 \left (\text {K\4089}^2+1\right )}\right \},\{y(x),\text {K\4089}\}\right ],\text {Solve}\left [\left \{y(x)=\frac {\sqrt {-a \text {K\4644}^2 \left (a \text {K\4644}^2-2 \text {K\4644}^2 x-2 x\right )}-a \text {K\4644}}{\text {K\4644}^2+1},x=\frac {a \text {K\4644}^2+a \text {K\4644}^2 \log ^2(\text {K\4644})+a \text {K\4644}^2 \log ^2\left (\sqrt {\text {K\4644}^2+1}+1\right )+a \log ^2\left (\sqrt {\text {K\4644}^2+1}+1\right )-2 a \text {K\4644}^2 \log (\text {K\4644}) \log \left (\sqrt {\text {K\4644}^2+1}+1\right )+2 a \sqrt {\text {K\4644}^2+1} \log (\text {K\4644})-2 a \log (\text {K\4644}) \log \left (\sqrt {\text {K\4644}^2+1}+1\right )-2 a \sqrt {\text {K\4644}^2+1} \log \left (\sqrt {\text {K\4644}^2+1}+1\right )+a c_1{}^2 \text {K\4644}^2-2 a c_1 \sqrt {\text {K\4644}^2+1}-2 a c_1 \text {K\4644}^2 \log (\text {K\4644})+2 a c_1 \text {K\4644}^2 \log \left (\sqrt {\text {K\4644}^2+1}+1\right )+2 a c_1 \log \left (\sqrt {\text {K\4644}^2+1}+1\right )+a \log ^2(\text {K\4644})-2 a c_1 \log (\text {K\4644})+a+a c_1{}^2}{2 \left (\text {K\4644}^2+1\right )}\right \},\{y(x),\text {K\4644}\}\right ]\right \}$ Maple : cpu = 1.255 (sec), leaf count = 111

$\left \{ [x \left ( {\it \_T} \right ) ={\frac {1}{2\,a} \left ( \left ( {\it Artanh} \left ( {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}} \right ) \right ) ^{2}\sqrt {{{\it \_T}}^{2}+1}{a}^{2}+ \left ( -2\,a{\it \_C1}\,\sqrt {{{\it \_T}}^{2}+1}-2\,{a}^{2} \right ) {\it Artanh} \left ( {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}} \right ) + \left ( {{\it \_C1}}^{2}+{a}^{2} \right ) \sqrt {{{\it \_T}}^{2}+1}+2\,{\it \_C1}\,a \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}},y \left ( {\it \_T} \right ) =-{{\it \_T} \left ( a{\it Artanh} \left ( {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}} \right ) -{\it \_C1} \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}+1}}}}] \right \}$