Solve the differential equation $e^{x}y\frac{dy}{dx}=e^{-y}+e^{-5x}-y$ by separation of variables.

• A. $y=e^{-5x}+C$
• B. $e^y=e^{-5x}+C$
• C. $\frac{y^2}{2}=e^{-5x}+C$
• D. $\frac{y^2}{2}+e^y=-\frac{e^{-5x}}{5}+C$

Answer: (D)

Verified step by step guidance

1. Step 1: Start by analyzing the given differential equation: \( e^x y \frac{dy}{dx} = e^{-y} + e^{-5x} - y \). The goal is to separate the variables \( x \) and \( y \) to solve the equation.
2. Step 2: Rewrite the equation to isolate \( \frac{dy}{dx} \): \( \frac{dy}{dx} = \frac{e^{-y} + e^{-5x} - y}{e^x y} \). This step prepares the equation for separation of variables.
3. Step 3: Attempt to separate the variables \( x \) and \( y \). Group terms involving \( y \) on one side and terms involving \( x \) on the other side. This may involve algebraic manipulation and factoring.
4. Step 4: Integrate both sides of the equation. For the \( y \)-dependent side, integrate with respect to \( y \). For the \( x \)-dependent side, integrate with respect to \( x \). Use appropriate integration techniques for exponential functions and polynomials.
5. Step 5: Combine the results of the integration and simplify to express the solution in the form \( \frac{y^2}{2} + e^y = -\frac{e^{-5x}}{5} + C \), where \( C \) is the constant of integration.