A Differential Equation is an equation that relates one or more [[Function|Functions]] and their [[Derivative|Derivatives]]. >[!example] Deriving Rate of Growth >The rate of growth of population $\deriv{p}{t}$ is proportional to its size, $\deriv{p}{t} \propto p(t)$, with an initial population of $p_0$, $p(0) = p_{0}$. > >$\huge >\begin{align} >\let p_{0} &= p(0)\\ > >\deriv{p}{t} &= rp \\ > >\frac{\d p}{p} &= r\d t \\ > >\int p^{-1} \d p &= \int r \d t \\ > \ln|p| &= rt + C \\ > \ln|p| &= rt+C \\ > > e^{\ln|p|} &= e^{rt+C} \\ >|p| &= e^{rt+C} \\ >\let A &= e^c\\ >p &= Ae^{rt} \\ > > >p(0) &= p_{0} = Ae^{0} \\ >p_{0} &= A \\ > >\\ > >p &= \boxed{p_{0}e^{rt}} >\end{align} >$