Solve the differential equation $e^{x}y\frac{dy}{dx}=e^{-y}+e^{-4x}-y$ by separation of variables.

• A. $y=\ln (C+e^{-4x})$
• B. $y=\ln (C-e^{-4x})$
• C. $y=-\ln (C-e^{-4x})$
• D. $y=C{e}^{-4x}$

Answer: (B)

Verified step by step guidance

1. Step 1: Start by analyzing the given differential equation: $e^x\; y\; \backslash (\backslash frac\{dy\}\{dx\}\backslash )\; =\; e^\{-y\}\; +\; e^\{-4x\}\; -\; y$. The goal is to separate the variables $x$ and $y$ to solve the equation.
2. Step 2: Rewrite the equation to isolate $\backslash (\backslash frac\{dy\}\{dx\}\backslash )$. Divide through by $e^x\; y$ to get: $\backslash (\backslash frac\{dy\}\{dx\}\backslash )\; =\; \backslash (\backslash frac\{e^\{-y\}\backslash )\; +\; e^\{-4x\}\; -\; y\}\{e^x\; y\}$.
3. Step 3: Check if the equation can be separated into terms involving only $y$ and terms involving only $x$. This involves algebraic manipulation to express the equation in the form $f(y)dy\; =\; g(x)dx$.
4. Step 4: Once the variables are separated, integrate both sides. For the $y$ side, integrate $f(y)$ with respect to $y$. For the $x$ side, integrate $g(x)$ with respect to $x$. Remember to include the constant of integration $C$.
5. Step 5: Solve for $y$ explicitly if possible. Use algebraic manipulation and properties of logarithms or exponentials to express $y$ in terms of $x$ and the constant $C$. Verify the solution by substituting back into the original differential equation.