Question

The integrating factor of the differential equation \( \frac{dy}{dx} + y = \frac{1 + y}{x} \) is:

Hint:

To solve linear first-order differential equations, find the integrating factor by using the formula \( \mu(x) = e^{\int P(x) \, dx} \).

Answer 1

The provided differential equation is: \[ \frac{dy}{dx} + y = \frac{1 + y}{x}. \] This can be rewritten as: \[ \frac{dy}{dx} + y = \frac{1}{x} + \frac{y}{x}. \] Rearranging the terms yields: \[ \frac{dy}{dx} + \frac{y}{x} = \frac{1}{x}. \] This equation is in the standard form of a linear first-order differential equation, \( \frac{dy}{dx} + P(x) y = Q(x) \), with \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{1}{x} \). The integrating factor \( \mu(x) \) is calculated as: \[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln x} = x. \] The integrating factor is therefore \( \boxed{x} \).