Let y = f (x) be the solution to the initial value problem
Use Euler's method to estimate f (4) using 5 steps.
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
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 ynew and store the full value to your calculator, type
+ 1 → Y
Your calculator will add 1 (also known as yold) 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
(2.8 + Y) ÷ (2.8 - Y) × 0.4 =
and you get
This goes in the next box of the table, correct to a few decimal places
+ Y → Y.
You should get something along the lines of
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.