Let *y* = *f* (*x*) be the solution to the initial value problem

Use Euler's method to estimate *f* (4) using 5 steps.

Answer

To get from 2 to 4 in 5 steps means each step must have size

Starting off isn't so bad:

Now it gets a little complicated, so we'll walk through a couple of steps. Calculate

by typing the following into your calculator:

(2.4 + 1) ÷ (2.4 – 1) × 0.4

Your calculator should show something that looks like

0.9714285714.

In the table, write down the calculation and your answer to the first 4 decimal places or so. Your calculator will remember the rest of the decimal places for you.

To get *y*_{new} and store the full value to your calculator, type

+ 1 → *Y*

Your calculator will add 1 (also known as *y*_{old}) to 0.9714285714 (also known as Δ y) and store the result to *Y*. Write the calculation and your answer in the table, correct to the first 4 decimal places or so.

For the next step, calculate

by typing

(2.8 + *Y*) ÷ (2.8 – *Y*) × 0.4 =

and you get

2.303448276.

This goes in the next box of the table, correct to a few decimal places

Now type

+ *Y* → *Y*.

You should get something along the lines of

4.274878647.

Again, write part of this in the table and let the calculator remember the rest:

For the next row, we type

(3.2 + *Y*) ÷ (3.2 –* Y*) × 0.4 =

to get slope × Δ *x*, then

+ *Y* → *Y*

to update the *y* value.

One more time, and here's the final table:

We conclude *f* (4) ≈ 2.4602.

For the sake of comparison, here's what might happen if you didn't store the full numbers to your calculator:

This might not seem that far off from 2.4602, but when we're estimating we need all the accuracy we can get. Also, it's far enough from 2.4602 that you might not get full credit for that answer on an exam.