Theorem:

If P,Q are continuous on an open interval, $I$, containing $x_0$, then the initial value problem (IVP) $\frac{dy}{dx}+P(x)y=Q(x)$, $y(x_0)=y_0$ has a unique solution $y(x)$ on $I$ given by $y(x)=e^{-\int P(x)dx} \left(\displaystyle\int e^{\int P(x)dx}Q(x)dx + C \right)$ for an appropriate C