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

Concept: A first-order linear differential equation is of the form: \[ \frac{dy}{dx} + Py = Q \] Integrating Factor (I.F.) is: \[ \text{I.F.} = e^{\int P dx} \]

Step 1: Compare with standard form. \[ P = 1, \quad Q = e^{-x} \] \[ \text{I.F.} = e^{\int 1\,dx} = e^x \]

Step 2: Multiply the equation by I.F. \[ e^x \frac{dy}{dx} + e^x y = 1 \] \[ \frac{d}{dx}(ye^x) = 1 \]

Step 3: Integrate both sides. \[ \int d(ye^x) = \int 1\,dx \] \[ ye^x = x + C \]

Step 4: Find the general solution. \[ y = e^{-x}(x + C) \]