2.459   ODE No. 459

1. Problem in Latex
2. Mathematica input
3. Maple input

$$-{(y^{\prime }(x)-1)}^2+e^{-2x}{y}^{\prime }(x{)}^2+e^{-2y(x)}=0$$ ✓ Mathematica : cpu = 3.25094 (sec), leaf count = 271

$$\{\{y(x)\to \log (-\frac{e^{-c_1}(e^x+1)(-e^x+e^{x+2c_1}-1-e^{2c_1})}{\sqrt{8e^x+4e^{2x}+4}})\},\{y(x)\to \log \left(\frac{e^{-c_1}(e^x+1)(-e^x+e^{x+2c_1}-1-e^{2c_1})}{\sqrt{8e^x+4e^{2x}+4}}\right)\},\text{Solve}[-\frac{1}{2}\log (1-e^{2y(x)})+\frac{1}{2}\log (\sqrt{e^{2y(x)+2x}(e^{2y(x)}+e^{2x}-1)}+e^{2y(x)+x}-e^x-e^{2x})+\frac{1}{2}\log (\sqrt{e^{2y(x)+2x}(e^{2y(x)}+e^{2x}-1)}+e^{2y(x)+x}-e^x+e^{2x})-x-\frac{1}{2}\log (1-e^x)-\frac{1}{2}\log (e^x-1)=c_1,y(x)]\}$$ ✓ Maple : cpu = 0.625 (sec), leaf count = 65

$$\{y\left(x\right)=c_1+x+\ln \left((-1-\sqrt{-{\mathrm{e}}^{-2c_1}{\mathrm{e}}^{2x}+{\mathrm{e}}^{2x}}){\mathrm{e}}^{-x}\right),y\left(x\right)=c_1+x+\ln \left((-1+\sqrt{-{\mathrm{e}}^{-2c_1}{\mathrm{e}}^{2x}+{\mathrm{e}}^{2x}}){\mathrm{e}}^{-x}\right)\}$$