Question

Solution of the differential equation \[ \frac{dy}{dx} + 2y = e^{-x} \text{ is:} \]

• A. \( y e^x = x + c \)
• B. \( y e^{2x} = x + c \)
• C. \( y e^x = e^{2x} + c \)
• D. \( y e^{2x} = e^x + c \)

Hint:

For linear first-order differential equations, using the integrating factor method is an effective way to solve them.

Answer 1

The Correct Option is D

Step 1: Solve the differential equation.
We are given the differential equation: \[ \frac{dy}{dx} + 2y = e^{-x} \] This is a linear first-order differential equation. To solve it, we use the integrating factor method. The integrating factor is \( e^{\int 2dx} = e^{2x} \). Step 2: Multiply through by the integrating factor.
Multiply the entire equation by \( e^{2x} \): \[ e^{2x} \frac{dy}{dx} + 2y e^{2x} = e^{2x} e^{-x} \] Simplifying: \[ \frac{d}{dx} \left( y e^{2x} \right) = e^{x} \] Step 3: Integrate both sides.
Now, integrate both sides with respect to \( x \): \[ \int \frac{d}{dx} \left( y e^{2x} \right) dx = \int e^x dx \] \[ y e^{2x} = e^x + c \] Step 4: Conclusion.
Thus, the solution is \( \boxed{y e^{2x} = e^x + c} \).