A linear first-order differential equation has the form:

\[ \frac{dy}{dx} + P(x)y = Q(x) \]

where \(y\) is the dependent variable, \(x\) is the independent variable, and \(P(x)\) and \(Q(x)\) are known functions of \(x\). To solve this differential equation, we can use an integrating factor. The integrating factor, denoted by \( \mu(x) \), is defined as:

\[ \mu(x) = e^{\int P(x) \,dx} \]

Multiplying both sides of the differential equation by the integrating factor, we get:

\[ e^{\int P(x) \,dx} \frac{dy}{dx} + P(x)e^{\int P(x) \,dx}y = Q(x)e^{\int P(x) \,dx} \]

This expression can be rewritten using the product rule of differentiation:

\[ \frac{d}{dx} \left( e^{\int P(x) \,dx}y \right) = Q(x)e^{\int P(x) \,dx} \]

Now, integrate both sides with respect to \(x\):

\[ \int \frac{d}{dx} \left( e^{\int P(x) \,dx}y \right) \,dx = \int Q(x)e^{\int P(x) \,dx} \,dx \]

The left side can be simplified using the reverse chain rule:

\[ e^{\int P(x) \,dx}y = \int Q(x)e^{\int P(x) \,dx} \,dx + C