Let y be the solution to the differential equation
that passes through the point (0, 0). Use Euler's Method to estimate y(1) with step size Δ x = 0.25.
There will be 4 steps, involving lines at x = 0, x = 0.25, x = 0.5, and x = 0.75
First Step: We start at the point (0, 0) and draw a line to estimate y(0.25). The slope of this line is
Second Step: We start at the point (0.25,0) so that yold = 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 yold = 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 yold = 0.375 and draw a line to estimate y(1). The slope of this line is
Our final estimate, using Euler's method, is that
y(1) ≈ 0.75.
The function y = f (x) is a solution to the differential equation
and f (0) = 5. Estimate y(1) using Euler's method with four steps.
To get from 0 to 1 in four steps means
Here's the final table:
We conclude that
f (1) ≈ 4.9247
The function y = f(x) is a solution to the d.e. y ' = x + y.
Estimate f(2) using Euler's method with 4 steps given that
f(0) = 1.
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.
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.