Table of Contents
Differential Equations
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Topic PlaylistThis may seem like a silly problem made up for a maths course, but it was actually studied in depth by Bernoulli and other mathematicians. Bernoulli investigated what happens to a savings account with interest when the compounding occurs infinitely often. The problem results in a formula for Euler's number $e$ and can be written as a differential equation as the the number of compoundings goes to infinity.
Explanation VideoStaring principal $p$, the initial amount in the savings account, and an interest rate $r$ compound $n$ times per year, the calculation for the total amount in the savings account at the end of the year is,
\begin{equation} p \left( 1 + \frac{r}{n} \right)^{n} \end{equation} Solution VideoCompare the different amounts in a savings account after one year with with initial principal of 100 and an interest rate of 5% compound yearly, monthly, and daily.
Explanation VideoApproximate the amount in a savings account after one year with with initial principal of 1 and an interest rate of 5% compound continuously. Compare the value with $e^{r}$. Set the interest rate to 100% to calculate an approximate value for $e$.
\begin{equation} \lim_{n \rightarrow \infty} \left( 1 + \frac{r}{n} \right)^{n} \end{equation} Explanation VideoWith compound interest $n$ times per year, define $\Delta t = \dfrac{1}{n}$ to create the equation
\begin{equation} S(t + \Delta t) = S(t)(1+\Delta t r) + \Delta t k \end{equation}The limit as $\Delta t \rightarrow 0$ results in the ODE,
\begin{equation} \frac{\text{d}S}{\text{d}t} = rS+k \end{equation}where $r$ is the interest rate and $k$ is the amount deposited per year.
Solution VideoIf you save for retirement for 40 years in savings account at 5% interest, how much should you deposit each month to have 1,000,000 when you retire?