Suppose you were asked to solve the initial value problem below using Euler's Method with step size 0.1 . What would be the value of $y[2]$ ? Note: the 2 should be a subscript.
\[
\frac{d y}{d x}=x+y
\]
\[
y(0)=2
\]
a. 2.1
b. 2.2
c. 2.33
d. 2.43

Step 1 ：The problem is asking to solve the given differential equation using Euler's method with a step size of 0.1. Euler's method is a numerical method used to solve first order first degree differential equation with a given initial value.

Step 2 ：The formula for Euler's method is: \(y_{n+1} = y_n + h*f(x_n, y_n)\) where h is the step size, f(x, y) is the derivative of y with respect to x, and (x_n, y_n) is the current point.

Step 3 ：In this case, the function f(x, y) is given by the differential equation dy/dx = x + y. The initial value y(0) is given as 2. We need to find the value of y at x = 0.2 (since we are taking steps of 0.1 and we need to find the value of y[2], which corresponds to the second step).

Step 4 ：Using the formula for Euler's method, we calculate the value of y at x = 0.2.

Step 5 ：The result is 2.43, which matches option d in the original question. Therefore, the final answer is 2.43.

Step 6 ：Final Answer: The value of \(y[2]\) is \(\boxed{2.43}\).