Question

Pick the CORRECT solution for the following differential equation:
\[ \frac{dy}{dx} = e^{x - y} \]

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

Hint:

When solving differential equations, always separate variables first, integrate both sides, and apply the necessary logarithmic operations for solutions involving exponential terms.

Answer 1

The Correct Option is A

The given differential equation is: \[ \frac{dy}{dx} = e^{x - y} \] Step 1: Rearranging the equation: \[ \frac{dy}{dx} = e^x \cdot e^{-y} \] Step 2: Separating the variables: \[ e^y \, dy = e^x \, dx \] Step 3: Integrating both sides: \[ \int e^y \, dy = \int e^x \, dx \] Step 4: Performing the integrations: \[ e^y = e^x + C \] Step 5: Taking the natural logarithm of both sides: \[ y = \ln(e^x + C) \] Thus, the correct solution is \( y = \ln(e^x + C) \), which corresponds to option (A).