Question

Find the general solution of the differential equation \( \frac{dy}{dx} + y = e^{-x} \).

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

Hint:

For linear differential equations, always compute the integrating factor first—it simplifies the equation into an exact derivative.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The given equation is a first-order Linear Differential Equation. We need to find the relation between \( y \) and \( x \).
Step 2: Key Formula or Approach:
For \( \frac{dy}{dx} + Py = Q \), the Integrating Factor (I.F.) is \( e^{\int P \, dx} \).
The general solution is \( y \cdot (I.F.) = \int Q \cdot (I.F.) \, dx + C \).
Step 3: Detailed Explanation:
Comparing \( \frac{dy}{dx} + y = e^{-x} \) with the standard form, we get:
\( P = 1 \) and \( Q = e^{-x} \).
Calculating I.F.:
\[ I.F. = e^{\int 1 \, dx} = e^x \]
Applying the general solution formula:
\[ y \cdot e^x = \int e^{-x} \cdot e^x \, dx + C \]
\[ y \cdot e^x = \int e^{0} \, dx + C \]
\[ y \cdot e^x = \int 1 \, dx + C \]
\[ y \cdot e^x = x + C \]
Step 4: Final Answer:
Dividing both sides by \( e^x \) or multiplying by \( e^{-x} \):
\[ y = e^{-x}(x + C) \]