1 Problems 2,4 (a) (5.3) from the textbook[10 points] Use Taylor's method of order two and four to approximate the solution for the following initial- value problem with h = 0.5. y' (t ) = e(t -u(t) te [0, 1], y (0) = 1. 2 Problem 2 (d) (5.4) from the textbook[10 points] Use the Modified Euler method with h = 0.25 to approximate the solution of following initial-value problems and compare the result to the actual value. y' (t) = (sin(2t) - 2ty(t)) te [1, 2], y(1) = 2. Actual value y(t) = 4+ cos(2) - cos(2t) 2t2 3 Problem 4 (c) (5.4) from the textbook[10 [points] Use the Modified Euler method with h = 0.2 to approximate the solution to the following initial- value problem and compare the result to the actual values. [y'(t) = 4(02 +U(0 te [ 1, 3], ly(1 ) = -2. Actual value y (t) = 2t 1 - 2t 4 Problem 18 (c) (5.4) from the textbook[10 points] Use the results of and linear interpolation to approximate values of y(1.3) and y(2.93). 5 Taylor's Polynomial [10 points] Find first degree Taylor polynomials for the following two-variable functions: (a) F(x, y) = erty +2x + y about (0, 0), (b) G(x, y) = sin(x - y) +x2 + (y - 1)2 about (0, 0). 6 Problem 20 (5.6) from the textbook[15 points] Derive Milne's method by applying the open Newton-Cotes formula (4.29) to the integral y ( ti+ 1 ) - y(ti-3 ) = / f (t, y (t ) ) at .1 Problems 2,4 (a) (5.3) from the textbook[10 points] Use Taylor's method of order two and four to approximate the solution for the following initial- value problem with h = 0.5. y' (t ) = e(t -u(t) te [0, 1], y (0) = 1. 2 Problem 2 (d) (5.4) from the textbook[10 points] Use the Modified Euler method with h = 0.25 to approximate the solution of following initial-value problems and compare the result to the actual value. y' (t) = (sin(2t) - 2ty(t)) te [1, 2], y(1) = 2. Actual value y(t) = 4+ cos(2) - cos(2t) 2t2 3 Problem 4 (c) (5.4) from the textbook[10 [points] Use the Modified Euler method with h = 0.2 to approximate the solution to the following initial- value problem and compare the result to the actual values. [y'(t) = 4(02 +U(0 te [ 1, 3], ly(1 ) = -2. Actual value y (t) = 2t 1 - 2t 4 Problem 18 (c) (5.4) from the textbook[10 points] Use the results of and linear interpolation to approximate values of y(1.3) and y(2.93). 5 Taylor's Polynomial [10 points] Find first degree Taylor polynomials for the following two-variable functions: (a) F(x, y) = erty +2x + y about (0, 0), (b) G(x, y) = sin(x - y) +x2 + (y - 1)2 about (0, 0). 6 Problem 20 (5.6) from the textbook[15 points] Derive Milne's method by applying the open Newton-Cotes formula (4.29) to the integral y ( ti+ 1 ) - y(ti-3 ) = / f (t, y (t ) ) at