Use euler's method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem y' = x2y − 1 2 y2, y(0) = 9. (round your answer to four decimal places.) y(1) = need ?

Euler's method is a numerical method used in calculus to approximate a particular solution of a differential equation. As a numerical method, we have to apply the same procedure many times, until get the desired result.

In first place, we need to know all the values the problem is giving:

So, if the initial x-value is 0, and the step size is 0.2, the following x-value would be: ; then ; ; and so on.

Now, we have to apply the formula to find each y-value until get the match of , because the problem asks the solution y(1).

According to the Euler's method:

Where , and ; .

Replacing all values we calculate the y-value assigned to :

.

Now, , . We repeat the process with the new values:

Then, we repeat the same process until get the y-value for , which is , round to four decimal places.

Therefore, .

89

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