#### 2.390   ODE No. 390

$a y(x) y'(x)-b x-c+y'(x)^2=0$ Mathematica : cpu = 1.48113 (sec), leaf count = 142

$\text {Solve}\left [\left \{x=e^{b \left (\frac {\log (\text {K\306586})}{b}-\frac {\log \left (b-a \text {K\306586}^2\right )}{2 b}\right )} \left (\frac {\tan ^{-1}\left (\frac {\sqrt {a} \text {K\306586}}{\sqrt {b-a \text {K\306586}^2}}\right )}{\sqrt {a}}-\frac {c \sqrt {b-a \text {K\306586}^2}}{b \text {K\306586}}\right )+c_1 e^{b \left (\frac {\log (\text {K\306586})}{b}-\frac {\log \left (b-a \text {K\306586}^2\right )}{2 b}\right )},y(x)=\frac {b x}{a \text {K\306586}}+\frac {c-\text {K\306586}^2}{a \text {K\306586}}\right \},\{y(x),\text {K\306586}\}\right ]$ Maple : cpu = 1.32 (sec), leaf count = 281

$\left \{ y \left ( x \right ) =2\,{\frac {{{\rm e}^{{\it RootOf} \left ( \sqrt {a}{\it \_C1}\,b{{\rm e}^{2\,{\it \_Z}}}-a{{\rm e}^{2\,{\it \_Z}}}bx+\sqrt {a}{\it \_C1}\,{b}^{2}-{{\rm e}^{2\,{\it \_Z}}}{\it \_Z}\,b-a{{\rm e}^{2\,{\it \_Z}}}c+a{b}^{2}x-{\it \_Z}\,{b}^{2}+abc \right ) }} \left ( -1/4\, \left ( {{\rm e}^{2\,{\it RootOf} \left ( \sqrt {a}{\it \_C1}\,b{{\rm e}^{2\,{\it \_Z}}}-a{{\rm e}^{2\,{\it \_Z}}}bx+\sqrt {a}{\it \_C1}\,{b}^{2}-{{\rm e}^{2\,{\it \_Z}}}{\it \_Z}\,b-a{{\rm e}^{2\,{\it \_Z}}}c+a{b}^{2}x-{\it \_Z}\,{b}^{2}+abc \right ) }}+b \right ) ^{2}{{\rm e}^{-2\,{\it RootOf} \left ( \sqrt {a}{\it \_C1}\,b{{\rm e}^{2\,{\it \_Z}}}-a{{\rm e}^{2\,{\it \_Z}}}bx+\sqrt {a}{\it \_C1}\,{b}^{2}-{{\rm e}^{2\,{\it \_Z}}}{\it \_Z}\,b-a{{\rm e}^{2\,{\it \_Z}}}c+a{b}^{2}x-{\it \_Z}\,{b}^{2}+abc \right ) }}+a \left ( bx+c \right ) \right ) }{{a}^{3/2} \left ( {{\rm e}^{2\,{\it RootOf} \left ( \sqrt {a}{\it \_C1}\,b{{\rm e}^{2\,{\it \_Z}}}-a{{\rm e}^{2\,{\it \_Z}}}bx+\sqrt {a}{\it \_C1}\,{b}^{2}-{{\rm e}^{2\,{\it \_Z}}}{\it \_Z}\,b-a{{\rm e}^{2\,{\it \_Z}}}c+a{b}^{2}x-{\it \_Z}\,{b}^{2}+abc \right ) }}+b \right ) }} \right \}$