0$, $N_0>P$ and the population is initially more than $P$ then decays towards $P$.}\\ % \text{Divergent:}\quad&\text{When $k>0,$\quad$e^{kt}$ is getting larger and approaches infinity when $t\to+\infty$.}\\ &\text{If $A<0$, $N_0
0$, $N_0>P$ and the population is initially more than $P$ then grows away from $P$ towards $+\infty$.}\\ \\ \text{Derived Use:}\quad&\text{When a problem is in the form of $\frac{dN}{dt}$ proportional to $N\pm C$ (a constant), the same can apply.}\\ &\text{e.g. For a falling object in air, }\frac{dv}{dt}=g-kv\:,\quad\frac{dv}{dt}=-k\left(v-\frac{g}{k}\right)\:,\quad\text{one can derived that}\\ &v=\frac{g}{k}+\left(v_0-\frac{g}{k}\right)e^{-kt}\:.\quad \text{If the object was initially at rest, $v_0=0$ and }v=\frac{g}{k}\left(1-e^{-kt}\right)\:.\\ % \text{Newton's Law of Cooling}\quad&\text{A system hotter than its surrounding will lose heat and gradually cool down towards an equilibrium}\\ &\text{temperature, which is assumed to be the surrounding temperature $T_s$ . So it is a case of}\\ &\text{convergence ($k<0$) with initial temperature higher than the surrounding ($T_0>T_s$ or $A>0$).}\\ &\frac{dT}{dt}=k(T-T_s)\:,\quad\boxed{T=T_s+A\:e^{kt}\:,\quad\text{where $k<0$ and $A=T_0-T_s>0$.}}\\ &\text{If the surrounding matter has a limited heat capacity (e.g. in a heat isolated tank), its}\\ &\text{temperature will follow the same transformation, with the same $T_s$ and $k$, but a different}\\ &\text{$T_0$ and therefore different $A$ (where $A<0$ as the surrounding is heating up towards $T_s$).}\\ &\text{(The above discussion can apply to a cooler system in a hotter room, with the sign of $A$ negated.)}\\ % \text{Brine:}\quad&\text{Given a volume $V$ of solution containing quantity $Q$ of a certain substance, every unit of}\\ &\text{time $v$ unit of solution with $q$ of substance was let in, fully mixed, then let out, find $Q(t)$.}\\ &\text{Every unit of time, the system is gaining $q$ but losing $Q\cdot\frac{v}{V}$, so $\frac{dQ}{dt}=q-kQ$, where $k=\frac{v}{V}$.}\\ &\frac{dQ}{dt}=-k\left(Q-\frac{\:q\:}{k}\right)\:,\quad Q=\frac{\:q\:}{k}+A\:e^{-kt}\:,\quad\text{where }A=Q_0-\frac{\:q\:}{k}\quad\text{and}\quad k=\frac{v}{V}\\ &Q=\frac{qV}{v}+\left(Q_0-\frac{qV}{v}\right)\:e^{-\frac{vt}{V}} =V\left[\frac{q}{v}+\left(\frac{Q_0}{V}-\frac{q}{v}\right)\:e^{-kt}\right] =V\left[\rho+\left(\rho_0-\rho\right)\:e^{-kt}\right]\\ &\boxed{Q=V\left[\rho+\left(\rho_0-\rho\right)\:e^{-kt}\right]}\\ &\text{where $\rho_0$ is the initial concentration, $\rho$ is that of the ``refreshment'', and $k$ is the ``refreshment'' ratio.}\\ &\text{The function always converges, becoming more concentrated if $\rho_0<\rho$, i.e. $A<0$.}\\ \text{Generally,}\quad&\boxed{Q=\frac{\:q\:}{k}+\left(Q_0-\frac{\:q\:}{k}\right)\:e^{-kt}}\\ &\text{if there are $q$ ``new borns'' every unit of time, and $k$ is the ``death rate'' ($0