Given the initial-value problem y = 1 t2 y t y2 , 1 t 2, y(1) = 1, with exact solution y(t) = 1/t: a. Use Taylors method of order two with h = 0.05 to approximate the solution, and compare it with the actual values of y. b. Use the answers generated in part (a) and linear interpolation to approximate the following values of y, and compare them to the actual values. i. y(1.052) ii. y(1.555) iii. y(1.978) c. Use Taylors method of order four with h = 0.05 to approximate the solution, and compare it with the actual values of y. d. Use the answers generated in part (c) and piecewise cubic Hermite interpolation to approximate the following values of y, and compare them to the actual values. i. y(1.052) ii. y(1.555) iii. y(1.978)

MATH 340 – INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS What We Covered: April 4 1. Course Content – Chapter 9: Linear Systems with Constant Coefficients a. Section 9.6: The Exponential of a Matrix 1 2 i. The exponential of the matrix A is defined to be = 2! + + 1 3 ∞ 1 3! +...= ∑=0! ii. Proposition: Suppose A is an nxn matrix 1. Then =