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 given differential equation is: \[ \frac{dy}{dx} + y = \frac{1 + y}{x}. \] Rewriting it: \[ \frac{dy}{dx} + y = \frac{1}{x} + \frac{y}{x}. \] Now, rearranging the terms: \[ \frac{dy}{dx} + \frac{y}{x} = \frac{1}{x}. \] This is a linear first-order differential equation in the form \( \frac{dy}{dx} + P(x) y = Q(x) \), where \( P(x) = \frac{1}{x} \) and \( Q(x) = \frac{1}{x} \). The integrating factor \( \mu(x) \) is given by: \[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln x} = x. \] Thus, the integrating factor is \( \boxed{x} \).