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To solve the differential equation use the integrating factor $\mu(x) = e^{\int P(x) dx}$, which transforms the equation into an exact differential.

For $P(x) = p$, a constant, the solution is $y = \frac{1}{e^{px}}\left(\int e^{px}Q(x) dx + C\right)$.

The general solution is $y = \frac{1}{\mu(x)}\left(\int \mu(x)g(x) dx + C\right)$ where $\mu(x)$ is the integrating factor.

Initial conditions allow for the determination of the constant of integration $C$, thus providing the particular solution.

Once the integrating factor is applied, the resulting equation is separable, leading to a solution that can be expressed as an integral.

Both the integrating factor technique and the Laplace Transform method are used to solve linear differential equations, but Laplace Transforms are particularly useful for equations with discontinuous or impulsive forcing functions.

Once a first-order linear differential equation is made exact using an integrating factor, the general solution can be found by integrating the exact equation.

The equation for charging an RC circuit, $\frac{dq}{dt} + \frac{1}{RC}q = E$, can be solved using an integrating factor, yielding the charge $q(t)$ at any time $t$.

For $Q(x) = 0$, the general solution is $y = Ce^{-\int P(x) dx}$, where $C$ is the constant of integration.