#### 2.403   ODE No. 403

$a y'(x)^2+b y'(x)-y(x)=0$ Mathematica : cpu = 0.227671 (sec), leaf count = 116

$\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {\#1} a+b^2}+b \log \left (b-\sqrt {4 \text {\#1} a+b^2}\right )}{2 a}\& \right ]\left [\frac {x}{2 a}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {\#1} a+b^2}-b \log \left (\sqrt {4 \text {\#1} a+b^2}+b\right )}{2 a}\& \right ]\left [c_1-\frac {x}{2 a}\right ]\right \}\right \}$ Maple : cpu = 3.888 (sec), leaf count = 197

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