For the systemthe curve Y(t) = (cos t, sin t) is a solution. This solution is periodic. It...

For the system

the curve Y(t) = (cos t, sin t) is a solution. This solution is periodic. Its initial position is Y(0) = (1, 0), and it returns to this position when t = 2π. So Y(2π) = (1, 0) and Y(t + 2π) = Y(t) for all t.

(a) Check that Y(t) = (cos t, sin t) is a solution.

(b) Use Euler’s method with step size 0.5 to approximate this solution, and check how close the approximate solution is to the real solution when t = 4, t = 6, and t = 10.

(c) Use Euler’s method with step size 0.1 to approximate this solution, and check how close the approximate solution is to the real solution when t = 4, t = 6, and t = 10.

(d) The points on the solution curve Y(t) are all 1 unit distance from the origin. Is this true of the approximate solutions? Are they too far from the origin or too close to it? What will happen for other step sizes (that is, will approximate solutions formed with other step sizes be too far or too close to the origin)?

[Use a computer or calculator to perform Euler’s method.]