Question

Consider the initial value problem y′=sin(πt),y(1)=3

Use Euler's Method with five steps to approximate y(2) to six decimal places (do not round intermediate results).
y(2)≈ ___

Answer 1

To approximate y(2), we can use Euler's Method with five steps to calculate y2.

Step-by-step explanation:

To approximate y(2) using Euler's Method, we first need to calculate the step size, h. In this case, h = (2-1)/5 = 0.2. Next, we initialize the values of t and y. Since y(1) = 3, we set t1 = 1 and y1 = 3. Then, we can use the Euler's Method formula: yn+1 = yn + h*f(tn, yn), where f(t, y) = sin(πt).

Plugging in the values, we get y2 ≈ 3 + 0.2*sin(π*1) ≈ 3 + 0.2*sin(π) ≈ 3 + 0.2*0 ≈ 3.

Therefore, y(2) ≈ 3 to six decimal places.