Differential Equations Exercises

Answer

First Step: We start at the point (0, 0) and draw a line to estimate y(0.25). The slope of this line is

We have

Second Step: We start at the point (0.25,0) so that y_old = 0 and draw a line to estimate y(.5). The slope of this line is

Then we can estimate

Third Step: We start at the point (0.5, 0.125) so that y_old = 0.125 and draw a line to estimate y(0.75). The slope of this line is

Then we estimate

Fourth Step: We start at the point (0.75,0.375) so that y_old = 0.375 and draw a line to estimate y(1). The slope of this line is

Then

Our final estimate, using Euler's method, is that

y(1) ≈ 0.75.

Answer

To get from 0 to 1 in four steps means

Here's the final table:

We conclude that

f (1) ≈ 4.9247

Answer

Here's the final table:

We can conclude that

f (2) ≈ 7.125.

When doing Euler's method, after each step you can save the new value of y to your calculator. That way you don't need to retype all the decimal places to calculate the next value of y, and you don't need to worry about rounding until the end of the problem. Write down numbers to 3 or 4 decimal places in the table, but use your calculator so that all the decimal places get used to calculate each value of y.

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.