2.430   ODE No. 430

$\text {a0} x+y'(x) (\text {a1} x+\text {b1} y(x)+\text {c1})+(\text {a2} x+\text {c2}) y'(x)^2+\text {b0} y(x)+\text {c0}=0$ Mathematica : cpu = 245.198 (sec), leaf count = 478

$\text {Solve}\left [\left \{x=(\text {b0}+\text {b1} \text {K\363169}) \exp \left (\frac {(\text {b1} (\text {b0}-\text {a1})+2 \text {a2} \text {b0}) \tan ^{-1}\left (\frac {\text {a1}+2 \text {K\363169} (\text {a2}+\text {b1})+\text {b0}}{\sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}\right )}{(\text {a2}+\text {b1}) \sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}-\frac {(2 \text {a2}+\text {b1}) \log (\text {a0}+\text {K\363169} (\text {a1}+\text {K\363169} (\text {a2}+\text {b1})+\text {b0}))}{2 (\text {a2}+\text {b1})}\right ) \int \frac {\left (\frac {-\text {c1}-2 \text {c2} \text {K\363169}}{\text {b0}+\text {b1} \text {K\363169}}-\frac {\text {b1} \left (-\text {c0}-\text {c1} \text {K\363169}-\text {c2} \text {K\363169}^2\right )}{(\text {b0}+\text {b1} \text {K\363169})^2}\right ) \exp \left (\frac {(2 \text {a2}+\text {b1}) \log (\text {a0}+\text {K\363169} (\text {a1}+\text {K\363169} (\text {a2}+\text {b1})+\text {b0}))}{2 (\text {a2}+\text {b1})}-\frac {(\text {b1} (\text {b0}-\text {a1})+2 \text {a2} \text {b0}) \tan ^{-1}\left (\frac {\text {a1}+2 \text {K\363169} (\text {a2}+\text {b1})+\text {b0}}{\sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}\right )}{(\text {a2}+\text {b1}) \sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}\right )}{(\text {b0}+\text {b1} \text {K\363169}) \left (\text {K\363169}-\frac {-\text {a0}-\text {a1} \text {K\363169}-\text {a2} \text {K\363169}^2}{\text {b0}+\text {b1} \text {K\363169}}\right )} \, d\text {K\363169}+c_1 (\text {b0}+\text {b1} \text {K\363169}) \exp \left (\frac {(\text {b1} (\text {b0}-\text {a1})+2 \text {a2} \text {b0}) \tan ^{-1}\left (\frac {\text {a1}+2 \text {K\363169} (\text {a2}+\text {b1})+\text {b0}}{\sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}\right )}{(\text {a2}+\text {b1}) \sqrt {4 \text {a0} (\text {a2}+\text {b1})-\text {a1}^2-2 \text {a1} \text {b0}-\text {b0}^2}}-\frac {(2 \text {a2}+\text {b1}) \log (\text {a0}+\text {K\363169} (\text {a1}+\text {K\363169} (\text {a2}+\text {b1})+\text {b0}))}{2 (\text {a2}+\text {b1})}\right ),y(x)=\frac {x \left (-\text {a0}-\text {a1} \text {K\363169}-\text {a2} \text {K\363169}^2\right )}{\text {b0}+\text {b1} \text {K\363169}}+\frac {-\text {c0}-\text {c1} \text {K\363169}-\text {c2} \text {K\363169}^2}{\text {b0}+\text {b1} \text {K\363169}}\right \},\{y(x),\text {K\363169}\}\right ]$ Maple : cpu = 1.983 (sec), leaf count = 1602

$\left \{ {\frac {1}{2\,{\it a2}\,x+2\,{\it c2}} \left ( -2\, \left ( {\it \_C1}-1/2\,\int ^{{\frac {-{\it a1}\,x-{\it b1}\,y \left ( x \right ) -{\it c1}+\sqrt {{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it b1}\,{\it a1}-4\,{\it a2}\,{\it b0} \right ) x-4\,{\it b0}\,{\it c2}+2\,{\it b1}\,{\it c1} \right ) y \left ( x \right ) + \left ( -4\,{\it a0}\,{\it a2}+{{\it a1}}^{2} \right ) {x}^{2}+ \left ( -4\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1}-4\,{\it a2}\,{\it c0} \right ) x-4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}}{2\,{\it a2}\,x+2\,{\it c2}}}}\!-{\frac { \left ( 2\,{\it \_a}\,{\it c2}+{\it c1} \right ) {\it b0}+{\it b1}\, \left ( {{\it \_a}}^{2}{\it c2}-{\it c0} \right ) }{ \left ( {\it \_a}\,{\it b1}+{\it b0} \right ) ^{2}}{{\rm e}^{{\frac {-2\,{\it a2}\,{\it b0}+{\it b1}\, \left ( {\it a1}-{\it b0} \right ) }{\sqrt { \left ( 4\,{\it a2}+4\,{\it b1} \right ) {\it a0}- \left ( {\it a1}+{\it b0} \right ) ^{2}} \left ( {\it a2}+{\it b1} \right ) }\arctan \left ( {\frac { \left ( 2\,{\it a2}+2\,{\it b1} \right ) {\it \_a}+{\it a1}+{\it b0}}{\sqrt { \left ( 4\,{\it a2}+4\,{\it b1} \right ) {\it a0}- \left ( {\it a1}+{\it b0} \right ) ^{2}}}} \right ) }}} \left ( \left ( {\it a2}+{\it b1} \right ) {{\it \_a}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_a}+{\it a0} \right ) ^{-{\frac {{\it b1}}{2\,{\it a2}+2\,{\it b1}}}}}{d{\it \_a}} \right ) \left ( -{\it b1}\,\sqrt {{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it b1}\,{\it a1}-4\,{\it a2}\,{\it b0} \right ) x-4\,{\it b0}\,{\it c2}+2\,{\it b1}\,{\it c1} \right ) y \left ( x \right ) + \left ( -4\,{\it a0}\,{\it a2}+{{\it a1}}^{2} \right ) {x}^{2}+ \left ( -4\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1}-4\,{\it a2}\,{\it c0} \right ) x-4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}+{{\it b1}}^{2}y \left ( x \right ) + \left ( {\it a1}\,x+{\it c1} \right ) {\it b1}-2\,{\it b0}\, \left ( {\it a2}\,x+{\it c2} \right ) \right ) {{\rm e}^{{\frac {-2\,{\it a2}\,{\it b0}+{\it b1}\, \left ( {\it a1}-{\it b0} \right ) }{\sqrt { \left ( 4\,{\it a2}+4\,{\it b1} \right ) {\it a0}- \left ( {\it a1}+{\it b0} \right ) ^{2}} \left ( {\it a2}+{\it b1} \right ) }\arctan \left ( {\frac { \left ( -{\it a2}-{\it b1} \right ) \sqrt {{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it b1}\,{\it a1}-4\,{\it a2}\,{\it b0} \right ) x-4\,{\it b0}\,{\it c2}+2\,{\it b1}\,{\it c1} \right ) y \left ( x \right ) + \left ( -4\,{\it a0}\,{\it a2}+{{\it a1}}^{2} \right ) {x}^{2}+ \left ( -4\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1}-4\,{\it a2}\,{\it c0} \right ) x-4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}+{\it b1}\, \left ( {\it a2}+{\it b1} \right ) y \left ( x \right ) + \left ( -{\it b0}\,x+{\it c1} \right ) {\it a2}+ \left ( {\it a1}\,x+{\it c1} \right ) {\it b1}-{\it c2}\, \left ( {\it a1}+{\it b0} \right ) }{\sqrt { \left ( 4\,{\it a2}+4\,{\it b1} \right ) {\it a0}- \left ( {\it a1}+{\it b0} \right ) ^{2}} \left ( {\it a2}\,x+{\it c2} \right ) }} \right ) }}} \left ( 1/2\,{\frac { \left ( -{\it b1}\, \left ( {\it a2}+{\it b1} \right ) y \left ( x \right ) + \left ( -{\it a1}\,x-{\it c1} \right ) {\it b1}+ \left ( {\it b0}\,x-{\it c1} \right ) {\it a2}+{\it c2}\, \left ( {\it a1}+{\it b0} \right ) \right ) \sqrt {{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it b1}\,{\it a1}-4\,{\it a2}\,{\it b0} \right ) x-4\,{\it b0}\,{\it c2}+2\,{\it b1}\,{\it c1} \right ) y \left ( x \right ) + \left ( -4\,{\it a0}\,{\it a2}+{{\it a1}}^{2} \right ) {x}^{2}+ \left ( -4\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1}-4\,{\it a2}\,{\it c0} \right ) x-4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}+{{\it b1}}^{2} \left ( {\it a2}+{\it b1} \right ) \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it a1}\,x+2\,{\it c1} \right ) {{\it b1}}^{2}+ \left ( \left ( \left ( {\it a1}-3\,{\it b0} \right ) x+2\,{\it c1} \right ) {\it a2}-{\it c2}\, \left ( {\it a1}+3\,{\it b0} \right ) \right ) {\it b1}-2\,{\it a2}\,{\it b0}\, \left ( {\it a2}\,x+{\it c2} \right ) \right ) y \left ( x \right ) + \left ( \left ( -2\,{\it a0}\,{x}^{2}-2\,{\it c0}\,x \right ) {\it a2}+{{\it a1}}^{2}{x}^{2}+ \left ( -2\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1} \right ) x-2\,{\it c0}\,{\it c2}+{{\it c1}}^{2} \right ) {\it b1}-2\,{{\it a2}}^{2}{\it c0}\,x+ \left ( -{\it a1}\,{\it b0}\,{x}^{2}+ \left ( 2\,{\it c2}\,{\it a0}+{\it c1}\, \left ( {\it a1}-{\it b0} \right ) \right ) x-2\,{\it c0}\,{\it c2}+{{\it c1}}^{2} \right ) {\it a2}-{\it c2}\, \left ( {\it a1}\, \left ( {\it a1}+{\it b0} \right ) x-2\,{\it c2}\,{\it a0}+{\it c1}\, \left ( {\it a1}+{\it b0} \right ) \right ) }{ \left ( {\it a2}\,x+{\it c2} \right ) ^{2}}} \right ) ^{{\frac {-2\,{\it a2}-{\it b1}}{2\,{\it a2}+2\,{\it b1}}}}+2\,{\it a2}\,{x}^{2}+2\,{\it c2}\,x \right ) }=0,{\frac {1}{2\,{\it a2}\,x+2\,{\it c2}} \left ( 2\, \left ( {\it \_C1}+1/2\,\int ^{{\frac {-{\it a1}\,x-{\it b1}\,y \left ( x \right ) -\sqrt {{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it b1}\,{\it a1}-4\,{\it a2}\,{\it b0} \right ) x-4\,{\it b0}\,{\it c2}+2\,{\it b1}\,{\it c1} \right ) y \left ( x \right ) + \left ( -4\,{\it a0}\,{\it a2}+{{\it a1}}^{2} \right ) {x}^{2}+ \left ( -4\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1}-4\,{\it a2}\,{\it c0} \right ) x-4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}-{\it c1}}{2\,{\it a2}\,x+2\,{\it c2}}}}\!-{\frac { \left ( 2\,{\it \_a}\,{\it c2}+{\it c1} \right ) {\it b0}+{\it b1}\, \left ( {{\it \_a}}^{2}{\it c2}-{\it c0} \right ) }{ \left ( {\it \_a}\,{\it b1}+{\it b0} \right ) ^{2}}{{\rm e}^{{\frac {-2\,{\it a2}\,{\it b0}+{\it b1}\, \left ( {\it a1}-{\it b0} \right ) }{\sqrt { \left ( 4\,{\it a2}+4\,{\it b1} \right ) {\it a0}- \left ( {\it a1}+{\it b0} \right ) ^{2}} \left ( {\it a2}+{\it b1} \right ) }\arctan \left ( {\frac { \left ( 2\,{\it a2}+2\,{\it b1} \right ) {\it \_a}+{\it a1}+{\it b0}}{\sqrt { \left ( 4\,{\it a2}+4\,{\it b1} \right ) {\it a0}- \left ( {\it a1}+{\it b0} \right ) ^{2}}}} \right ) }}} \left ( \left ( {\it a2}+{\it b1} \right ) {{\it \_a}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_a}+{\it a0} \right ) ^{-{\frac {{\it b1}}{2\,{\it a2}+2\,{\it b1}}}}}{d{\it \_a}} \right ) \left ( {\it b1}\,\sqrt {{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it b1}\,{\it a1}-4\,{\it a2}\,{\it b0} \right ) x-4\,{\it b0}\,{\it c2}+2\,{\it b1}\,{\it c1} \right ) y \left ( x \right ) + \left ( -4\,{\it a0}\,{\it a2}+{{\it a1}}^{2} \right ) {x}^{2}+ \left ( -4\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1}-4\,{\it a2}\,{\it c0} \right ) x-4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}+{{\it b1}}^{2}y \left ( x \right ) + \left ( {\it a1}\,x+{\it c1} \right ) {\it b1}-2\,{\it b0}\, \left ( {\it a2}\,x+{\it c2} \right ) \right ) {{\rm e}^{{\frac {-2\,{\it a2}\,{\it b0}+{\it b1}\, \left ( {\it a1}-{\it b0} \right ) }{\sqrt { \left ( 4\,{\it a2}+4\,{\it b1} \right ) {\it a0}- \left ( {\it a1}+{\it b0} \right ) ^{2}} \left ( {\it a2}+{\it b1} \right ) }\arctan \left ( {\frac { \left ( {\it a2}+{\it b1} \right ) \sqrt {{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it b1}\,{\it a1}-4\,{\it a2}\,{\it b0} \right ) x-4\,{\it b0}\,{\it c2}+2\,{\it b1}\,{\it c1} \right ) y \left ( x \right ) + \left ( -4\,{\it a0}\,{\it a2}+{{\it a1}}^{2} \right ) {x}^{2}+ \left ( -4\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1}-4\,{\it a2}\,{\it c0} \right ) x-4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}+{\it b1}\, \left ( {\it a2}+{\it b1} \right ) y \left ( x \right ) + \left ( -{\it b0}\,x+{\it c1} \right ) {\it a2}+ \left ( {\it a1}\,x+{\it c1} \right ) {\it b1}-{\it c2}\, \left ( {\it a1}+{\it b0} \right ) }{\sqrt { \left ( 4\,{\it a2}+4\,{\it b1} \right ) {\it a0}- \left ( {\it a1}+{\it b0} \right ) ^{2}} \left ( {\it a2}\,x+{\it c2} \right ) }} \right ) }}} \left ( 1/2\,{\frac { \left ( {\it b1}\, \left ( {\it a2}+{\it b1} \right ) y \left ( x \right ) + \left ( -{\it b0}\,x+{\it c1} \right ) {\it a2}+ \left ( {\it a1}\,x+{\it c1} \right ) {\it b1}-{\it c2}\, \left ( {\it a1}+{\it b0} \right ) \right ) \sqrt {{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it b1}\,{\it a1}-4\,{\it a2}\,{\it b0} \right ) x-4\,{\it b0}\,{\it c2}+2\,{\it b1}\,{\it c1} \right ) y \left ( x \right ) + \left ( -4\,{\it a0}\,{\it a2}+{{\it a1}}^{2} \right ) {x}^{2}+ \left ( -4\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1}-4\,{\it a2}\,{\it c0} \right ) x-4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}+{{\it b1}}^{2} \left ( {\it a2}+{\it b1} \right ) \left ( y \left ( x \right ) \right ) ^{2}+ \left ( \left ( 2\,{\it a1}\,x+2\,{\it c1} \right ) {{\it b1}}^{2}+ \left ( \left ( \left ( {\it a1}-3\,{\it b0} \right ) x+2\,{\it c1} \right ) {\it a2}-{\it c2}\, \left ( {\it a1}+3\,{\it b0} \right ) \right ) {\it b1}-2\,{\it a2}\,{\it b0}\, \left ( {\it a2}\,x+{\it c2} \right ) \right ) y \left ( x \right ) + \left ( \left ( -2\,{\it a0}\,{x}^{2}-2\,{\it c0}\,x \right ) {\it a2}+{{\it a1}}^{2}{x}^{2}+ \left ( -2\,{\it c2}\,{\it a0}+2\,{\it a1}\,{\it c1} \right ) x-2\,{\it c0}\,{\it c2}+{{\it c1}}^{2} \right ) {\it b1}-2\,{{\it a2}}^{2}{\it c0}\,x+ \left ( -{\it a1}\,{\it b0}\,{x}^{2}+ \left ( 2\,{\it c2}\,{\it a0}+{\it c1}\, \left ( {\it a1}-{\it b0} \right ) \right ) x-2\,{\it c0}\,{\it c2}+{{\it c1}}^{2} \right ) {\it a2}-{\it c2}\, \left ( {\it a1}\, \left ( {\it a1}+{\it b0} \right ) x-2\,{\it c2}\,{\it a0}+{\it c1}\, \left ( {\it a1}+{\it b0} \right ) \right ) }{ \left ( {\it a2}\,x+{\it c2} \right ) ^{2}}} \right ) ^{{\frac {-2\,{\it a2}-{\it b1}}{2\,{\it a2}+2\,{\it b1}}}}+2\,{\it a2}\,{x}^{2}+2\,{\it c2}\,x \right ) }=0 \right \}$