#### 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 = 7.37684 (sec), leaf count = 553

$\left \{\text {Solve}\left [\left \{y(x)=\frac {-\sqrt {-a \text {K\414258}^2 \left (a \text {K\414258}^2-2 \text {K\414258}^2 x-2 x\right )}-a \text {K\414258}}{\text {K\414258}^2+1},x=\frac {a c_1{}^2 \text {K\414258}^2-2 a c_1 \sqrt {\text {K\414258}^2+1}-2 a c_1 \text {K\414258}^2 \log (\text {K\414258})+2 a c_1 \text {K\414258}^2 \log \left (\sqrt {\text {K\414258}^2+1}+1\right )+2 a c_1 \log \left (\sqrt {\text {K\414258}^2+1}+1\right )-2 a c_1 \log (\text {K\414258})+a c_1{}^2+a \text {K\414258}^2+a \text {K\414258}^2 \log ^2(\text {K\414258})+a \text {K\414258}^2 \log ^2\left (\sqrt {\text {K\414258}^2+1}+1\right )+a \log ^2\left (\sqrt {\text {K\414258}^2+1}+1\right )-2 a \text {K\414258}^2 \log (\text {K\414258}) \log \left (\sqrt {\text {K\414258}^2+1}+1\right )+2 a \sqrt {\text {K\414258}^2+1} \log (\text {K\414258})-2 a \log (\text {K\414258}) \log \left (\sqrt {\text {K\414258}^2+1}+1\right )-2 a \sqrt {\text {K\414258}^2+1} \log \left (\sqrt {\text {K\414258}^2+1}+1\right )+a \log ^2(\text {K\414258})+a}{2 \left (\text {K\414258}^2+1\right )}\right \},\{y(x),\text {K\414258}\}\right ],\text {Solve}\left [\left \{y(x)=\frac {\sqrt {-a \text {K\414811}^2 \left (a \text {K\414811}^2-2 \text {K\414811}^2 x-2 x\right )}-a \text {K\414811}}{\text {K\414811}^2+1},x=\frac {a c_1{}^2 \text {K\414811}^2-2 a c_1 \sqrt {\text {K\414811}^2+1}-2 a c_1 \text {K\414811}^2 \log (\text {K\414811})+2 a c_1 \text {K\414811}^2 \log \left (\sqrt {\text {K\414811}^2+1}+1\right )+2 a c_1 \log \left (\sqrt {\text {K\414811}^2+1}+1\right )-2 a c_1 \log (\text {K\414811})+a c_1{}^2+a \text {K\414811}^2+a \text {K\414811}^2 \log ^2(\text {K\414811})+a \text {K\414811}^2 \log ^2\left (\sqrt {\text {K\414811}^2+1}+1\right )+a \log ^2\left (\sqrt {\text {K\414811}^2+1}+1\right )-2 a \text {K\414811}^2 \log (\text {K\414811}) \log \left (\sqrt {\text {K\414811}^2+1}+1\right )+2 a \sqrt {\text {K\414811}^2+1} \log (\text {K\414811})-2 a \log (\text {K\414811}) \log \left (\sqrt {\text {K\414811}^2+1}+1\right )-2 a \sqrt {\text {K\414811}^2+1} \log \left (\sqrt {\text {K\414811}^2+1}+1\right )+a \log ^2(\text {K\414811})+a}{2 \left (\text {K\414811}^2+1\right )}\right \},\{y(x),\text {K\414811}\}\right ]\right \}$ Maple : cpu = 1.194 (sec), leaf count = 111

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