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## Homework Statement

Consider the equation

y0 = Ly; y(0) = 1:

**L = lamda**

Verify that the solution to this equation is y(t) = e^(Lt). We want to solve this equation numerically to obtain an approximation to y(1). Consider the two following methods to approximate

the solution to this equation:

Method I: y_(n+1) = y_(n) + hf(y_(n+1); t_(n+1))

Method II:y_(n+1) = y_(n) +(h/2)*(f(y_(n); t_(n)) + f(y_(n+1); t_(n+1)))

(a) Compute the truncation error for both methods. Which one is more accurate?

(b) i was able to do this one!!

(c) Obtain an expression for y_(n+1) as a function of y0 for methods I and II.

(d) Using the expressions obtained in the previous part, compute lim (as n goes to inf) y_(n)

(Remember: h = 1/n).

(e) In view of the results in the previous part, do the methods converge?

(f) Assume that h = :1, and L = -1000. Plot y0, y1, y2, y3, and y4 for both

methods. Plot the function y(t) = e^(Lt). In view of these plots, which method performs

better?

i am just soo lost when it comes to solving this problem. i have idea how to approach it. any help with this will be greatly appreciated!