Solve by separation of variables e^{x}y*dy/dx=e^{-y} + e^{-2x-y}.

(1) $\begin{equation*} e^x y\frac{dy}{dx} = e^{-y} + e^{-2x-y} \end{equation*}$

We can separate the variables as follows:

(2) $\begin{equation*} e^y dy = \left(e^x + e^{-x}\right)dx \end{equation*}$

Now, we can integrate both sides of the equation:

(3) $\begin{equation*} \int e^y dy = \int \left(e^x + e^{-x}\right)dx \end{equation*}$

Integrating the left-hand side, we get:

(4) $\begin{equation*} e^y = x e^x - e^{-x} + C_1 \end{equation*}$

where $C_1$ is the constant of integration.

Simplifying and solving for $y$ , we have:

(5) $\begin{equation*} y = \ln\left(x e^x - e^{-x} + C\right) \end{equation*}$

where $C = e^{C_1}$ is a constant of integration.

Therefore, the solution to the differential equation $e^x y\frac{dy}{dx} = e^{-y} + e^{-2x-y}$ is:

(6) $\begin{equation*} y = \ln\left(x e^x - e^{-x} + C\right) \end{equation*}$

where $C$ is a constant of integration.