2.403   ODE No. 403

\[ a y'(x)^2+b y'(x)-y(x)=0 \] Mathematica : cpu = 0.224895 (sec), leaf count = 116


\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} a+b^2}+b \log \left (b-\sqrt {4 \text {$\#$1} a+b^2}\right )}{2 a}\& \right ]\left [\frac {x}{2 a}+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {\sqrt {4 \text {$\#$1} a+b^2}-b \log \left (\sqrt {4 \text {$\#$1} a+b^2}+b\right )}{2 a}\& \right ]\left [-\frac {x}{2 a}+c_1\right ]\right \}\right \}\] Maple : cpu = 1.182 (sec), leaf count = 247


\[y \relax (x ) = \frac {{\mathrm e}^{-\frac {b \ln \left (\frac {1}{4 a}\right )+2 b \LambertW \left (\frac {2 \,{\mathrm e}^{-\frac {c_{1}}{b}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{b}}}{b \sqrt {\frac {1}{a}}}\right )+2 c_{1}+2 b -2 x}{2 b}} \left ({\mathrm e}^{-\frac {b \ln \left (\frac {1}{4 a}\right )+2 b \LambertW \left (\frac {2 \,{\mathrm e}^{-\frac {c_{1}}{b}} {\mathrm e}^{-1} {\mathrm e}^{\frac {x}{b}}}{b \sqrt {\frac {1}{a}}}\right )+2 c_{1}+2 b -2 x}{2 b}}+2 b \right )}{4 a}\]