2.459   ODE No. 459

\[ -\left (y'(x)-1\right )^2+e^{-2 x} y'(x)^2+e^{-2 y(x)}=0 \]

✓ Mathematica : cpu = 13.7031 (sec), leaf count = 397

DSolve[E^(-2*y[x]) - (-1 + Derivative[1][y][x])^2 + Derivative[1][y][x]^2/E^(2*x) == 0,y[x],x]
 

\[\left \{\left \{y(x)\to \log \left (-\frac {i \left (e^x-1\right ) e^{c_1 \left (-e^{-x}\right ) \sqrt {e^{2 x}}} \left (e^x+e^{2 c_1 e^{-x} \sqrt {e^{2 x}}}+e^{x+2 c_1 e^{-x} \sqrt {e^{2 x}}}-1\right )}{\sqrt {-8 e^x+4 e^{2 x}+4}}\right )\right \},\left \{y(x)\to \log \left (\frac {i \left (e^x-1\right ) e^{c_1 \left (-e^{-x}\right ) \sqrt {e^{2 x}}} \left (e^x+e^{2 c_1 e^{-x} \sqrt {e^{2 x}}}+e^{x+2 c_1 e^{-x} \sqrt {e^{2 x}}}-1\right )}{\sqrt {-8 e^x+4 e^{2 x}+4}}\right )\right \},\left \{y(x)\to \log \left (-\frac {i \left (e^x+1\right ) e^{c_1 \left (-e^{-x}\right ) \sqrt {e^{2 x}}} \left (e^x+e^{2 c_1 e^{-x} \sqrt {e^{2 x}}}+e^{x+2 c_1 e^{-x} \sqrt {e^{2 x}}}-1\right )}{\sqrt {8 e^x+4 e^{2 x}+4}}\right )\right \},\left \{y(x)\to \log \left (\frac {i \left (e^x+1\right ) e^{c_1 \left (-e^{-x}\right ) \sqrt {e^{2 x}}} \left (e^x+e^{2 c_1 e^{-x} \sqrt {e^{2 x}}}+e^{x+2 c_1 e^{-x} \sqrt {e^{2 x}}}-1\right )}{\sqrt {8 e^x+4 e^{2 x}+4}}\right )\right \}\right \}\]

✓ Maple : cpu = 0.624 (sec), leaf count = 113

dsolve(exp(-2*x)*diff(y(x),x)^2-(diff(y(x),x)-1)^2+exp(-2*y(x)) = 0,y(x))
 

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