#### 2.501   ODE No. 501

$y'(x)^2 \left (a y(x)^2+b x+c\right )-b y(x) y'(x)+d y(x)^2=0$ Mathematica : cpu = 21.8209 (sec), leaf count = 913

$\left \{\text {Solve}\left [\left \{y(x)=\frac {b \text {K\446980}-\sqrt {-\text {K\446980}^2 \left (-b^2+4 a \text {K\446980}^2 x b+4 d x b+4 a c \text {K\446980}^2+4 c d\right )}}{2 \left (a \text {K\446980}^2+d\right )},x=\frac {-b^2 c_1{}^2 d^4-a b^2 \text {K\446980}^2 c_1{}^2 d^3+2 b^2 c_1 \log (\text {K\446980}) d^{5/2}-2 b^2 c_1 \log \left (d+\sqrt {a \text {K\446980}^2+d} \sqrt {d}\right ) d^{5/2}-4 c d^2+2 b^2 \sqrt {a \text {K\446980}^2+d} c_1 d^2+2 a b^2 \text {K\446980}^2 c_1 \log (\text {K\446980}) d^{3/2}-2 a b^2 \text {K\446980}^2 c_1 \log \left (d+\sqrt {a \text {K\446980}^2+d} \sqrt {d}\right ) d^{3/2}-4 a c \text {K\446980}^2 d-b^2 \log ^2(\text {K\446980}) d-b^2 \log ^2\left (d+\sqrt {a \text {K\446980}^2+d} \sqrt {d}\right ) d+2 b^2 \log (\text {K\446980}) \log \left (d+\sqrt {a \text {K\446980}^2+d} \sqrt {d}\right ) d-2 b^2 \sqrt {a \text {K\446980}^2+d} \log (\text {K\446980}) \sqrt {d}+2 b^2 \sqrt {a \text {K\446980}^2+d} \log \left (d+\sqrt {a \text {K\446980}^2+d} \sqrt {d}\right ) \sqrt {d}-a b^2 \text {K\446980}^2 \log ^2(\text {K\446980})-a b^2 \text {K\446980}^2 \log ^2\left (d+\sqrt {a \text {K\446980}^2+d} \sqrt {d}\right )+2 a b^2 \text {K\446980}^2 \log (\text {K\446980}) \log \left (d+\sqrt {a \text {K\446980}^2+d} \sqrt {d}\right )}{4 b d \left (a \text {K\446980}^2+d\right )}\right \},\{y(x),\text {K\446980}\}\right ],\text {Solve}\left [\left \{y(x)=\frac {b \text {K\446999}+\sqrt {-\text {K\446999}^2 \left (-b^2+4 a \text {K\446999}^2 x b+4 d x b+4 a c \text {K\446999}^2+4 c d\right )}}{2 \left (a \text {K\446999}^2+d\right )},x=\frac {-b^2 c_1{}^2 d^4-a b^2 \text {K\446999}^2 c_1{}^2 d^3+2 b^2 c_1 \log (\text {K\446999}) d^{5/2}-2 b^2 c_1 \log \left (d+\sqrt {a \text {K\446999}^2+d} \sqrt {d}\right ) d^{5/2}-4 c d^2+2 b^2 \sqrt {a \text {K\446999}^2+d} c_1 d^2+2 a b^2 \text {K\446999}^2 c_1 \log (\text {K\446999}) d^{3/2}-2 a b^2 \text {K\446999}^2 c_1 \log \left (d+\sqrt {a \text {K\446999}^2+d} \sqrt {d}\right ) d^{3/2}-4 a c \text {K\446999}^2 d-b^2 \log ^2(\text {K\446999}) d-b^2 \log ^2\left (d+\sqrt {a \text {K\446999}^2+d} \sqrt {d}\right ) d+2 b^2 \log (\text {K\446999}) \log \left (d+\sqrt {a \text {K\446999}^2+d} \sqrt {d}\right ) d-2 b^2 \sqrt {a \text {K\446999}^2+d} \log (\text {K\446999}) \sqrt {d}+2 b^2 \sqrt {a \text {K\446999}^2+d} \log \left (d+\sqrt {a \text {K\446999}^2+d} \sqrt {d}\right ) \sqrt {d}-a b^2 \text {K\446999}^2 \log ^2(\text {K\446999})-a b^2 \text {K\446999}^2 \log ^2\left (d+\sqrt {a \text {K\446999}^2+d} \sqrt {d}\right )+2 a b^2 \text {K\446999}^2 \log (\text {K\446999}) \log \left (d+\sqrt {a \text {K\446999}^2+d} \sqrt {d}\right )}{4 b d \left (a \text {K\446999}^2+d\right )}\right \},\{y(x),\text {K\446999}\}\right ]\right \}$ Maple : cpu = 5.392 (sec), leaf count = 215

$\left \{ [x \left ( {\it \_T} \right ) =-{\frac {1}{4\,bd} \left ( \left ( \ln \left ( {\frac {1}{{\it \_T}} \left ( \sqrt {d}\sqrt {{{\it \_T}}^{2}a+d}+d \right ) } \right ) \right ) ^{2}\sqrt {{{\it \_T}}^{2}a+d}{b}^{2}+ \left ( \left ( 2\,\ln \left ( 2 \right ) {b}^{2}+4\,\sqrt {d}{\it \_C1}\,b \right ) \sqrt {{{\it \_T}}^{2}a+d}-2\,\sqrt {d}{b}^{2} \right ) \ln \left ( {\frac {1}{{\it \_T}} \left ( \sqrt {d}\sqrt {{{\it \_T}}^{2}a+d}+d \right ) } \right ) + \left ( 4\,\ln \left ( 2 \right ) \sqrt {d}{\it \_C1}\,b+ \left ( \ln \left ( 2 \right ) \right ) ^{2}{b}^{2}+4\,d \left ( {{\it \_C1}}^{2}+c \right ) \right ) \sqrt {{{\it \_T}}^{2}a+d}-2\,\ln \left ( 2 \right ) \sqrt {d}{b}^{2}-4\,d{\it \_C1}\,b \right ) {\frac {1}{\sqrt {{{\it \_T}}^{2}a+d}}}},y \left ( {\it \_T} \right ) ={\frac {{\it \_T}}{2} \left ( b\ln \left ( {\frac {1}{{\it \_T}} \left ( \sqrt {d}\sqrt {{{\it \_T}}^{2}a+d}+d \right ) } \right ) +b\ln \left ( 2 \right ) +2\,{\it \_C1}\,\sqrt {d} \right ) {\frac {1}{\sqrt {d}}}{\frac {1}{\sqrt {{{\it \_T}}^{2}a+d}}}}] \right \}$