Answer:
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:
- The step size is 0.2; h = 0.2. This step size is a periodical increase of the x-variable, which will allow us to calculate each y-value to each x.
- The problem is asking the solution y(1), which means that we have to find the y-value assigned for x = 1, through the numerical method.
- The initial condition is y(0) = 9. In other words,
.
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, .