In the past two weeks, I continued my learning on First Order Linear Differential Equations. In this blog, I will focus on the Bank Account Model. If you find this blog is interesting and would like to learn more from my primary source, please go here.
Bank Account Model:
When you are putting money into your bank account, what would you care about the most? While I guess the answer for me and many will be the interest. So in this model, we will look at the logarithm behind the (theoretical) interest system.
Suppose that there are x dollars in my bank account based on a specific time t, we can easily derive the function x(t) as a representation of my balance at a specific time. Furthermore, also suppose right now that the bank pays me at a constant annual interest rate of r. From the assumptions, we can easily see that r is a constant, x is a function of t, and t is a variable with a constant rate of change.
A long time ago, banks calculated and paid interest once a month, which means that in this model t would no longer be a continuous variable. To find the monthly interest, we can assume Δt=1/12 and employ the following formula:
The function’s graph will look similar to y=[x], which means round down x to the largest integer.
In nowadays, however, interest is calculated based on much smaller intervals by computers. Therefore, we can assume . Since now time is considered as a continuous variable, the interest I receive will be continuous as well.
If we assume Q(t) represents the total cumulative deposit and q(t) represents the amount I deposit every year, we can find the following DE:
Additionally, from the definition of q(t), we know that the total amount of my deposit and withdrawals across a period of time is q(t)*Δt.
Therefore, the previous formula
Can be rewritten as
After subtracting x(t) and divide by Δt on both sides, the formula can be further rewritten as the following. Please keep in mind that we are now assuming Δt approaches 0.
This formula can be easily reduced to the following using the definition of the derivative, which is a fundamental concept in the field of Calculus.
Now if we consider the interest rate r is no longer a constant, but a variable that changes base on time, then r(t) represents the specific interest rate at a specific time t. In this case, we can derive the following DE in the standard form of
Now, here is the theoretical bank interest model. If you interested in how to solve this model with mathematical techniques, please try out this link.
In this blog, we see how easily and closely a real-life problem may relate to differential equations. With my current knowledge, some of these problems can already be solved mathematically. In my next several weeks of studying, I will restart the process of exploring more Math knowledge. After that, more interesting problems will be discussed here.
The Definition Of The Derivative. (2018, May 30). Retrieved September 21, 2018, from Pauls Oline Notes website: http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx
How Much Money Can a Bank Hold? [Image]. (n.d.). Retrieved from https://wonderopolis.org/wonder/how-much-money-can-a-bank-hold
Modeling by First Order Linear ODE’s. (2011, Fall). Retrieved September 21, 2018, from MITOPENCOURSEWARE website: https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-i-first-order-differential-equations/basic-de-and-separable-equations/MIT18_03SCF11_s1_6text.pdf
Step Function. (n.d.). Retrieved September 21, 2018, from Desmos website: https://www.desmos.com/calculator/oq4vwcjltg
Step Function [Illustration]. (n.d.). Retrieved from https://www.desmos.com/calculator/oq4vwcjltg
Arthur Mattuck, Haynes Miller, Jeremy Orloff, and John Lewis. 18.03SC Differential Equations. Fall 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.