[Solved] MA 322 Lab 6

1. Use Eulers method to approximate the solutions for each of the following initial-valueproblems.

, with h = 0.5

, with h = 0.5

(c) ^dy_dt = y + ty^1/2, 2 t 3, y(2) = 2, with h = 0.25

, with h = 0.25

2. The actual solutions to the initial-value problems in problem 1 are given here. Computethe actual error
3. Given the initial-value problem

with exact solution.

• Use Eulers method with h = 0.05 to approximate the solution, and compare it with the actual values of y(t).
• Use the answers generated in part (a) and linear interpolation to approximate the following values of y(t), and compare them to the actual values.

(I) y(1.052) (II) y(1.555) (III) y(1.978)

4. Let h > 0 and let x_j = x₀ + jh (j = 1,2,,n) be given nodes. Consider the initial value problem , with

for all x [x₀,x_n] and for all y.

1. Using error analysis of the Eulers method, show that there exists an h > 0 such that

for some (x_n1,x_n), where e_n = y(x_n)y_n with y_n obtained using Euler method. ii. Applying the conclusion of (i) above recursively, prove that

|e_n| |e₀| + nh²Y, where, Y . (1)

• The solution of the initial value problem

is y(x) = sinx. For = 20, find the approximate value of y(3) using the Eulers method with h = 0.5. Compute the error bound given in (1), and Show that the actual absolute error exceeds the computed error bound given in (1). Explain why it does not contradict the validity of (1).

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