#### 2.459   ODE No. 459

$-\left (y'(x)-1\right )^2+e^{-2 x} y'(x)^2+e^{-2 y(x)}=0$ Mathematica : cpu = 2.63538 (sec), leaf count = 271

$\left \{\left \{y(x)\to \log \left (-\frac {e^{-c_1} \left (e^x+1\right ) \left (-e^x+e^{x+2 c_1}-1-e^{2 c_1}\right )}{\sqrt {8 e^x+4 e^{2 x}+4}}\right )\right \},\left \{y(x)\to \log \left (\frac {e^{-c_1} \left (e^x+1\right ) \left (-e^x+e^{x+2 c_1}-1-e^{2 c_1}\right )}{\sqrt {8 e^x+4 e^{2 x}+4}}\right )\right \},\text {Solve}\left [-\frac {1}{2} \log \left (1-e^{2 y(x)}\right )+\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x-e^{2 x}\right )+\frac {1}{2} \log \left (\sqrt {e^{2 y(x)+2 x} \left (e^{2 y(x)}+e^{2 x}-1\right )}+e^{2 y(x)+x}-e^x+e^{2 x}\right )-x-\frac {1}{2} \log \left (1-e^x\right )-\frac {1}{2} \log \left (e^x-1\right )=c_1,y(x)\right ]\right \}$ Maple : cpu = 0.949 (sec), leaf count = 65

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