12.4 Euler's method in spreadsheets

We set up a spreadsheet as below. The first row contains labels.

We copy the bottom cells down through Row 42, and plot columns A and B.

You need to go to the left, as it were, by finding $t$-values that are decreasing. That means you'll use $\Delta t = -0.1$.

Since we're taking values of $t$ that are decreasing, we'll use $\Delta t = -0.1$. Our spreadsheet setup is below. The first row contains labels.

We copy the columns down to Row 12 and plot columns A and B.

You'll be branching out from $t=0$ in two directions. So, you can copy-paste twice: once with $\Delta t=0.2$, and once with $\Delta t = -0.2$.

We'll set up our spreadsheet twice. First, in the normal way, with initial value $y(0)=0$ and step size $\Delta t = 0.2$. Then, with the same initial value, but $\Delta t = -0.2$. The first collection of data will approximate $y(t)$ to the right of $t=0$; the second collection will approximate $y(t)$ to the left of $t=0$.

So, our spreadsheet looks something like this (note the omitted rows):

The plot looks something like this:

There's no need to try to find tricks – just interpet the labels in Row 1 at face value.

There are, theoretically, different ways one could interpret the spreadsheet. But, one approach is by far the most obvious: to interpret the labels at face value.

The values of $\Delta t$ must be of the form $\frac1n$, where $n$ is a whole number; otherwise, you won't get an approximation for $y(1)$ specifically.

You aren't given an error tolerance, so it's up to you to play around with step sizes until your answers seems relatively consistent.

You aren't given an error tolerance, so your actual answer may vary, but if you've done a good job of making your step size relatively small, then your approximation should be relatively close to the actual answer of $\frac1{10}-\frac1{3000}=0.099\overline{66}$.

This question is meant to get you thinking about how to use spreadsheets creatively, but the vagueness makes it unlikely to show up on an exam in exactly this form.

We'll start with a fairly usual setup. We'll keep the step size stored separately, so we can change it later. Instead of copying our values down the column until we find a particular $t$-value, we're looking for a particular $y$-value. Or, at least, the spot where the $y_k$ values change from negative to positive.

The first iteration of Euler's method with step size 0.1 approximates $y(0.1)>0$. So, this approximation puts the root somewhere between 0 and 0.1 (but remember, it might not be entirely accurate).

Next, we will increase our accuracy by decreasing our step size. Changing the value in D2 to 0.1, and copying further down the columns, shows us $y(0.09) \approx-0.00961562$ and $y(0.1)\approx0.000385304$. So, this approximation guesses the root is between $0.09$ and $0.1$. We continue to decrease $\Delta t$ and note the $t$-values where the corresponding approximation changes from negative to positive.

There's not a clear stopping point, but the last approximation required about 1000 rows of the spreadsheet to produce, which seems on the edge of what is reasonable. So, our guess for the root is, let's say, $0.09965$.