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 = 2.69934 (sec), leaf count = 272

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

✓ Maple : cpu = 0.635 (sec), leaf count = 65

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