Use Euler's method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem given below. (Round your answer to four decimal places.)
y' = 1 − xy
y(0) = 0
I don't even know how to start!
you might start by reading up on Euler's method. google can help. There's even a youtube video illustrating the method.
There's a good discussion with examples here:
http://tutorial.math.lamar.edu/Classes/DE/EulersMethod.aspx
To use Euler's method, you need to follow these steps:
Step 1: Write the given initial-value problem in the form dy/dx = f(x, y), where f(x, y) represents the right-hand side of the differential equation.
In this case, we are given:
y' = 1 - xy
Step 2: Choose a step size, h. In this case, the step size is given as 0.2.
Step 3: Start with the initial condition, y(0) = 0.
Step 4: Apply the Euler's method formula to estimate y at the desired value of x.
The Euler's method formula is:
y_(n+1) = y_n + h * f(x_n, y_n)
where:
- y_(n+1) is the estimate of y at the next step
- y_n is the estimate of y at the current step
- h is the step size
- f(x_n, y_n) is the value of f(x, y) at the current step
Step 5: Repeat step 4 until you reach the desired value of x.
Now, let's apply these steps to estimate y(1) using Euler's method with a step size of 0.2.
We start with the given initial condition: y(0) = 0.
Step 4: Calculate the estimate of y at the next step:
y_(n+1) = y_n + h * (1 - x_n * y_n)
At the first step, with x = 0 and y = 0, we have:
y_1 = 0 + 0.2 * (1 - 0 * 0) = 0 + 0.2 * 1 = 0.2
Step 5: Repeat step 4 until reaching the desired value of x.
Using the same formula for the second step, with x = 0.2 and y = 0.2:
y_2 = 0.2 + 0.2 * (1 - 0.2 * 0.2) = 0.2 + 0.2 * (1 - 0.04) = 0.2 + 0.2 * 0.96 ≈ 0.392
Continue this process until you reach x = 1 or the desired value of x:
y_3 ≈ 0.582
y_4 ≈ 0.764
y_5 ≈ 0.91
y_6 ≈ 1.014
Therefore, using Euler's method with a step size of 0.2, the estimated value of y(1) is approximately 1.014 (rounded to four decimal places).