Question

Integrating factor of \[ \frac{dy}{dx} - 2y = 2x - 3 \]

• A. \( e^{2x} \)
• B. \( e^{-2x} \)
• C. \( e^{3x} \)
• D. \( e^{-3x} \)

Hint:

To solve first-order linear differential equations, the integrating factor is found by exponentiating the integral of the coefficient of \( y \).

Answer 1

The Correct Option is B

The given equation is a first-order linear differential equation. To find the integrating factor, we use the formula for the integrating factor for a linear equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \), which is: \[ \mu(x) = e^{\int P(x) \, dx} \] For the equation \( \frac{dy}{dx} - 2y = 2x - 3 \), \( P(x) = -2 \). Therefore, the integrating factor is: \[ \mu(x) = e^{\int -2 \, dx} = e^{-2x} \] Thus, the integrating factor is \( e^{-2x} \).