Question

Integrating factor of differential equation \( \frac{dy}{dx} - y = \cos x \) is \( e^{-x} \).

Hint:

The integrating factor for a linear differential equation is found by exponentiating the integral of the coefficient of \( y \).

Answer 1

Step 1: Identifying the integrating factor.
The differential equation is: \[ \frac{dy}{dx} - y = \cos x. \] This is a first-order linear differential equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \), where \( P(x) = -1 \) and \( Q(x) = \cos x \). The integrating factor \( \mu(x) \) is given by: \[ \mu(x) = e^{\int P(x) \, dx}. \] Since \( P(x) = -1 \), we have: \[ \mu(x) = e^{\int -1 \, dx} = e^{-x}. \]

Step 2: Conclusion.
Thus, the integrating factor is \( e^{-x} \), which makes the statement true.