3.3: Shall we save? (or withdraw. I don't really care what you do with your money.)

Section 3.3 is entitled “Personal Finance.”

Here’s another handy application for differential equations!

So P(t) will be the monetary balance at some time t of our bank account (how do you expect us to pay for our amoeba projects?). Our account will pay us some interest at a rate r per year, and this interest is compounded continuously. Between our time t and some change in t, which we will denote as Δt + t, would be P(Δt + t) – P(t) = interested earned in that time Δt. Since we originally denoted r as interest per year, this means that the interest earned over our change in time Δt is approximately (but not necessarily equal to) rPΔt.

So, like last section, we’ll take the derivative of our function P(t) by using the limit quotient definition of the derivative, which would make our differential equation P^’(t) = rP.

This is a differential equation that is easily separable and solvable (remember from some time ago, we called these types of differential equations exponential equations). Then our solution will have the form

Well, we should remind ourselves that this is real life, where we don’t just leave money in accounts forever and ever. I mean, we have amoebas to count. So let’s look at a savings account with a balance P(t) and we withdraw a balance W every year. There’s still an interest rate r that pays us interest every year (still compounded continuously). So our interest we earn will be similar to our first equation, but this time we add in the effect of withdrawals:

We’re going to make the same assumption like last time and say that the interest earned over this time is approximately rPΔt. However, we want to include something about the withdrawals. So we defined W as the money we withdraw per some time interval, so in a time Δt, we would withdraw WΔt. Rewriting our equation a bit, we would then have

Our derivative would now be rP – W, and thus this would be our model dP/dt. Conversely, if we deposit some amount of money D per year, our equation would be P^’ = rP + D.

There’s a paragraph about keeping track of the dimensions of the quantities (i.e. units). It’s just giving you reasons as to why we multiplied what we did to get the answers we did. As long as you keep track of units on each variable or constant, then you’ll be all good.

For the most part, this was all the new material in this section. I’m going to work through an example (since a majority of application problems and pretty much all of the problems you will come across for personal finance will be word problems). I don’t blame you, though, if you stop reading after this sentence.

Something else we do in real life is saving money for something important in our lives.  That might be for college. That might be for a house or a car. That might be for amoebas. (Okay, that was my last amoeba reference, I promise.)

Okay, just as an example, say we want to save for something that isn’t single-celled, so we want to put $1000 dollars in a savings account every year for 10 years. We start with nothing and we find a great %5 interest rate for our account. Very briefly we said that if we were to deposit money into an account and not withdraw anything, our equation would be P^’ = rP + D. Our r in this case would be 0.05 (since 5% is a percentage and all) and our D would be 1 (if we were to put our units in thousands of dollars). This would make our equation

Back in section 2.4, we said this is the form for a linear equation. Solving this is pretty simple, but I’ll summarize what we should do very quickly: find an integrating factor, multiply our equation by that integrating factor, integrate both sides and solve for P(t). Easy enough?

So our initial condition would be when P(0) = 0, since we started with nothing. This would make our C a positive 20, and then our final equation would be

That was pretty straightforward, right? As long as the differential equation isn’t too difficult to solve, then we’ll be golden.

We can add layers onto the personal finance application, by deciding how we’re going to save the money and where exactly that money will be coming from and when we’ll start to save that money. One of the examples, for instance, talks about a fixed percentage of our salary that we would deposit straight into our account and not touch. We could also add a layer of difficulty to that fixed percentage by making that an equation that would change as our salaries would increase (assuming our salaries increase when we work). In that case, we would be able to enjoy what little money we had at first and then save more as our salaries increased.

I guess the real moral of this section would be that life is hard and stuff is expensive and you have to save money to afford expensive stuff in the future. Now you can approximate how much money you have to save in order to afford expensive stuff! Yay differential equations!

By the way, this is the last section I have to summarize from this chapter. This is sad, considering the next section is about electrical circuits and this directly relates to my life. I'm quite ahead of the section my lecture is on, so I might just do electrical circuits because I would like to spread the physics love. 

In any event, I'll be seeing you something in the new future in chapter 4. Who knew time could fly so fast when we're having fun?