10.2 Simple Solution using Integrating Factor

$$\begin{align*} \frac{dy}{dx} + y \cdot p(x) &= q(x) \\ y' + yp &= q \\ I &= \int pdx, \quad \frac{dI}{dx} = p \\ \frac{d}{dx}(ye^I) &= y'e^I + y e^I \frac{dI}{dx} \\ &= y'e^I + y e^I p \\ &= e^I(y' + yp) \\ &= e^I q \\ ye^I &= \int (e^I q) + C \quad with \; some \; constant \; C\\ y &= e^{-I} \left[ \int \left( e^I q \right) + C \right] \end{align*}$$

$$\begin{align*} \frac{dP}{dt} + K_{d}P &= A_0 + \frac{mK}{2.303}t \quad (6.44) \\ y &= P, \quad x = t \\ p &= K_d, \quad q = A_0 + \frac{mK}{2.303}t \\ I &= \int_0^t K_d dt = K_d t, \quad e^I = e^{K_d t} \\ P &= e^{-K_d t} \left[ \int e^{K_d t} \left( A_0 + \frac{mK}{2.303} t \right) dt + C \right] \end{align*}$$