#### 2.479   ODE No. 479

$\text {a0} x+y'(x) (\text {a1} x+\text {b1} y(x)+\text {c1})+y'(x)^2 (\text {a2} x+\text {b2} y(x)+\text {c2})+\text {b0} y(x)+\text {c0}=0$ Mathematica : cpu = 9.0486 (sec), leaf count = 552

$\text {Solve}\left [\left \{x=-\frac {-(\text {b0}+\text {K\401181} (\text {b1}+\text {b2} \text {K\401181})) \exp \left (\text {RootSum}\left [\text {\#1}^3 \text {b2}+\text {\#1}^2 \text {a2}+\text {\#1}^2 \text {b1}+\text {\#1} \text {a1}+\text {\#1} \text {b0}+\text {a0}\& ,\frac {\text {\#1}^2 \text {b2} \log (\text {K\401181}-\text {\#1})+\text {b0} \log (\text {K\401181}-\text {\#1})+\text {\#1} \text {b1} \log (\text {K\401181}-\text {\#1})}{3 \text {\#1}^2 \text {b2}+2 \text {\#1} \text {a2}+2 \text {\#1} \text {b1}+\text {a1}+\text {b0}}\& \right ]\right ) \left (\int _1^{\text {K\401181}}\frac {\exp \left (-\text {RootSum}\left [\text {b2} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {b1} \text {\#1}^2+\text {a1} \text {\#1}+\text {b0} \text {\#1}+\text {a0}\& ,\frac {\text {b2} \log (K[1]-\text {\#1}) \text {\#1}^2+\text {b1} \log (K[1]-\text {\#1}) \text {\#1}+\text {b0} \log (K[1]-\text {\#1})}{3 \text {b2} \text {\#1}^2+2 \text {a2} \text {\#1}+2 \text {b1} \text {\#1}+\text {a1}+\text {b0}}\& \right ]\right ) (-\text {c0}-K[1] (\text {c1}+\text {c2} K[1]))}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]+c_1\right )+\text {c0}+\text {c1} \text {K\401181}+\text {c2} \text {K\401181}^2}{\text {a0}+\text {K\401181} (\text {a1}+\text {K\401181} (\text {a2}+\text {b1}+\text {b2} \text {K\401181})+\text {b0})},y(x)=-\frac {(\text {a0}+\text {K\401181} (\text {a1}+\text {a2} \text {K\401181})) \exp \left (\text {RootSum}\left [\text {\#1}^3 \text {b2}+\text {\#1}^2 \text {a2}+\text {\#1}^2 \text {b1}+\text {\#1} \text {a1}+\text {\#1} \text {b0}+\text {a0}\& ,\frac {\text {\#1}^2 \text {b2} \log (\text {K\401181}-\text {\#1})+\text {b0} \log (\text {K\401181}-\text {\#1})+\text {\#1} \text {b1} \log (\text {K\401181}-\text {\#1})}{3 \text {\#1}^2 \text {b2}+2 \text {\#1} \text {a2}+2 \text {\#1} \text {b1}+\text {a1}+\text {b0}}\& \right ]\right ) \left (\int _1^{\text {K\401181}}\frac {\exp \left (-\text {RootSum}\left [\text {b2} \text {\#1}^3+\text {a2} \text {\#1}^2+\text {b1} \text {\#1}^2+\text {a1} \text {\#1}+\text {b0} \text {\#1}+\text {a0}\& ,\frac {\text {b2} \log (K[1]-\text {\#1}) \text {\#1}^2+\text {b1} \log (K[1]-\text {\#1}) \text {\#1}+\text {b0} \log (K[1]-\text {\#1})}{3 \text {b2} \text {\#1}^2+2 \text {a2} \text {\#1}+2 \text {b1} \text {\#1}+\text {a1}+\text {b0}}\& \right ]\right ) (-\text {c0}-K[1] (\text {c1}+\text {c2} K[1]))}{\text {a0}+K[1] (\text {a1}+\text {b0}+K[1] (\text {a2}+\text {b1}+\text {b2} K[1]))}dK[1]+c_1\right )+\text {K\401181} (\text {c0}+\text {K\401181} (\text {c1}+\text {c2} \text {K\401181}))}{\text {a0}+\text {K\401181} (\text {a1}+\text {K\401181} (\text {a2}+\text {b1}+\text {b2} \text {K\401181})+\text {b0})}\right \},\{y(x),\text {K\401181}\}\right ]$ Maple : cpu = 0.234 (sec), leaf count = 929

$\left \{ x-{{\rm e}^{\int ^{{\frac {1}{2\,{\it b2}\,y \left ( x \right ) +2\,{\it a2}\,x+2\,{\it c2}} \left ( -{\it a1}\,x-{\it b1}\,y \left ( x \right ) -{\it c1}+\sqrt {-4\,{\it a0}\,{\it a2}\,{x}^{2}-4\,{\it a0}\,{\it b2}\,xy \left ( x \right ) +{{\it a1}}^{2}{x}^{2}+2\,{\it a1}\,{\it b1}\,xy \left ( x \right ) -4\,{\it a2}\,{\it b0}\,xy \left ( x \right ) -4\,{\it b0}\,{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,{\it a0}\,{\it c2}\,x+2\,{\it a1}\,{\it c1}\,x-4\,{\it a2}\,{\it c0}\,x-4\,{\it b0}\,{\it c2}\,y \left ( x \right ) +2\,{\it b1}\,{\it c1}\,y \left ( x \right ) -4\,{\it b2}\,{\it c0}\,y \left ( x \right ) -4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}} \right ) }}\!{\frac { \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) {{\it \_a}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}-2\,{\it a2}\,{\it b0} \right ) {\it \_a}+{\it a0}\,{\it b1}-{\it a1}\,{\it b0}}{ \left ( {{\it \_a}}^{2}{\it b2}+{\it \_a}\,{\it b1}+{\it b0} \right ) \left ( {{\it \_a}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_a}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_a}+{\it a0} \right ) }}{d{\it \_a}}}} \left ( \int ^{{\frac {1}{2\,{\it b2}\,y \left ( x \right ) +2\,{\it a2}\,x+2\,{\it c2}} \left ( -{\it a1}\,x-{\it b1}\,y \left ( x \right ) -{\it c1}+\sqrt {-4\,{\it a0}\,{\it a2}\,{x}^{2}-4\,{\it a0}\,{\it b2}\,xy \left ( x \right ) +{{\it a1}}^{2}{x}^{2}+2\,{\it a1}\,{\it b1}\,xy \left ( x \right ) -4\,{\it a2}\,{\it b0}\,xy \left ( x \right ) -4\,{\it b0}\,{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,{\it a0}\,{\it c2}\,x+2\,{\it a1}\,{\it c1}\,x-4\,{\it a2}\,{\it c0}\,x-4\,{\it b0}\,{\it c2}\,y \left ( x \right ) +2\,{\it b1}\,{\it c1}\,y \left ( x \right ) -4\,{\it b2}\,{\it c0}\,y \left ( x \right ) -4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}} \right ) }}\!-{\frac {{{\it \_b}}^{2}{\it b1}\,{\it c2}-{{\it \_b}}^{2}{\it b2}\,{\it c1}+2\,{\it \_b}\,{\it b0}\,{\it c2}-2\,{\it \_b}\,{\it b2}\,{\it c0}+{\it b0}\,{\it c1}-{\it b1}\,{\it c0}}{ \left ( {{\it \_b}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_b}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_b}+{\it a0} \right ) \left ( {{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0} \right ) }{{\rm e}^{-\int \!{\frac { \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) {{\it \_b}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}-2\,{\it a2}\,{\it b0} \right ) {\it \_b}+{\it a0}\,{\it b1}-{\it a1}\,{\it b0}}{ \left ( {{\it \_b}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_b}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_b}+{\it a0} \right ) \left ( {{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0} \right ) }}\,{\rm d}{\it \_b}}}}{d{\it \_b}}+{\it \_C1} \right ) =0,x-{{\rm e}^{\int ^{{\frac {1}{2\,{\it b2}\,y \left ( x \right ) +2\,{\it a2}\,x+2\,{\it c2}} \left ( -{\it a1}\,x-{\it b1}\,y \left ( x \right ) -\sqrt {-4\,{\it a0}\,{\it a2}\,{x}^{2}-4\,{\it a0}\,{\it b2}\,xy \left ( x \right ) +{{\it a1}}^{2}{x}^{2}+2\,{\it a1}\,{\it b1}\,xy \left ( x \right ) -4\,{\it a2}\,{\it b0}\,xy \left ( x \right ) -4\,{\it b0}\,{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,{\it a0}\,{\it c2}\,x+2\,{\it a1}\,{\it c1}\,x-4\,{\it a2}\,{\it c0}\,x-4\,{\it b0}\,{\it c2}\,y \left ( x \right ) +2\,{\it b1}\,{\it c1}\,y \left ( x \right ) -4\,{\it b2}\,{\it c0}\,y \left ( x \right ) -4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}-{\it c1} \right ) }}\!{\frac { \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) {{\it \_a}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}-2\,{\it a2}\,{\it b0} \right ) {\it \_a}+{\it a0}\,{\it b1}-{\it a1}\,{\it b0}}{ \left ( {{\it \_a}}^{2}{\it b2}+{\it \_a}\,{\it b1}+{\it b0} \right ) \left ( {{\it \_a}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_a}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_a}+{\it a0} \right ) }}{d{\it \_a}}}} \left ( \int ^{{\frac {1}{2\,{\it b2}\,y \left ( x \right ) +2\,{\it a2}\,x+2\,{\it c2}} \left ( -{\it a1}\,x-{\it b1}\,y \left ( x \right ) -\sqrt {-4\,{\it a0}\,{\it a2}\,{x}^{2}-4\,{\it a0}\,{\it b2}\,xy \left ( x \right ) +{{\it a1}}^{2}{x}^{2}+2\,{\it a1}\,{\it b1}\,xy \left ( x \right ) -4\,{\it a2}\,{\it b0}\,xy \left ( x \right ) -4\,{\it b0}\,{\it b2}\, \left ( y \left ( x \right ) \right ) ^{2}+{{\it b1}}^{2} \left ( y \left ( x \right ) \right ) ^{2}-4\,{\it a0}\,{\it c2}\,x+2\,{\it a1}\,{\it c1}\,x-4\,{\it a2}\,{\it c0}\,x-4\,{\it b0}\,{\it c2}\,y \left ( x \right ) +2\,{\it b1}\,{\it c1}\,y \left ( x \right ) -4\,{\it b2}\,{\it c0}\,y \left ( x \right ) -4\,{\it c0}\,{\it c2}+{{\it c1}}^{2}}-{\it c1} \right ) }}\!-{\frac {{{\it \_b}}^{2}{\it b1}\,{\it c2}-{{\it \_b}}^{2}{\it b2}\,{\it c1}+2\,{\it \_b}\,{\it b0}\,{\it c2}-2\,{\it \_b}\,{\it b2}\,{\it c0}+{\it b0}\,{\it c1}-{\it b1}\,{\it c0}}{ \left ( {{\it \_b}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_b}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_b}+{\it a0} \right ) \left ( {{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0} \right ) }{{\rm e}^{-\int \!{\frac { \left ( {\it a1}\,{\it b2}-{\it a2}\,{\it b1} \right ) {{\it \_b}}^{2}+ \left ( 2\,{\it a0}\,{\it b2}-2\,{\it a2}\,{\it b0} \right ) {\it \_b}+{\it a0}\,{\it b1}-{\it a1}\,{\it b0}}{ \left ( {{\it \_b}}^{3}{\it b2}+ \left ( {\it a2}+{\it b1} \right ) {{\it \_b}}^{2}+ \left ( {\it a1}+{\it b0} \right ) {\it \_b}+{\it a0} \right ) \left ( {{\it \_b}}^{2}{\it b2}+{\it \_b}\,{\it b1}+{\it b0} \right ) }}\,{\rm d}{\it \_b}}}}{d{\it \_b}}+{\it \_C1} \right ) =0 \right \}$