Use Euler's method with step size 0.2 to estimate y(1.4), where y(x) is the solution of the initial-value problem below. Give your answer correct to 4 decimal places.

y' = x - xy
y(1) = 0
h = 0.2

Since I am at y(1) = 0 and not y(0) = 0 would I just do this twice? As in:

y1 = 0 + 0.2*(1.2 - (1.2)(0))
y1 = 0.24

y2 = 0.24 + 0.2*(1.4 - (1.4)(0.24))
y2 = 0.4528?

Am I starting this out or setting it up wrong? Please help. Thx

y(x=1.2) = y(x=1) + 0.2* y'(1)
= 0 + 0.2* (1 - 0) = 0.2

y(x=1.4) = y(x=1.2) + 0.2 (y'@x=1.2)
= 0.2 + 0.2(1.2 - 1.2*0.2)
= 0.2 + 0.2(0.96) = 0.3920

give me an anser

Well, your calculations for y1 and y2 seem to be correct. However, it seems like you are not considering the correct initial condition for Euler's method.

To use Euler's method, you need to start with the given initial condition, which is y(1) = 0. Therefore, you don't need to do the calculation twice.

Here's the correct calculation using Euler's method with a step size of 0.2:

First, find y(1.2):
y(1.2) = y(1) + 0.2 * y'(1)
= 0 + 0.2 * (1 - 1 * 0)
= 0.2

Now, find y(1.4):
y(1.4) = y(1.2) + 0.2 * y'(1.2)
= 0.2 + 0.2 * (1.2 - 1.2 * 0.2)
= 0.2 + 0.2 * 0.96
= 0.2 + 0.192
= 0.392

Therefore, the estimated value of y(1.4) using Euler's method with a step size of 0.2 is 0.3920.

Keep in mind that Euler's method is an approximation method, so the result may not be as accurate as other numerical methods. But hey, at least it's better than trying to approximate it with clown jokes! 😄

Your calculation for y(1.2) using Euler's method is correct:

y(1.2) = 0 + 0.2 * (1.2 - (1.2)(0)) = 0.24

However, to find y(1.4), you need to use the value of y(1.2) that you obtained. The correct calculation is:

y(1.4) = y(1.2) + 0.2 * (1.4 - (1.4)(0.24))
= 0.24 + 0.2 * (1.4 - 0.336)
= 0.24 + 0.2 * 1.064
= 0.24 + 0.2128
= 0.4528

So the correct value for y(1.4) using Euler's method with a step size of 0.2 is 0.4528, rounded to 4 decimal places.

To solve the initial-value problem using Euler's method, you need to follow a step-by-step process. Here's how you can use Euler's method with a step size of 0.2 to estimate y(1.4):

Step 1: Set up the initial conditions. In this case, given y(1) = 0.

Step 2: Calculate y'(x) using the given differential equation. In this case, y'(x) = x - x*y.

Step 3: Use Euler's method to estimate y(1.2):
y(1.2) = y(1) + h * y'(1)
= 0 + 0.2 * (1 - 1 * 0)
= 0.2

Step 4: Use Euler's method again to estimate y(1.4):
y(1.4) = y(1.2) + h * y'(1.2)
= 0.2 + 0.2 * (1.2 - 1.2 * 0.2)
= 0.2 + 0.2 * 0.96
= 0.2 + 0.192
= 0.392

So, the estimated value of y(1.4) using Euler's method with a step size of 0.2 is 0.3920, rounded to 4 decimal places.

Yes that is the right answer