Applied Mathematics I - Fall 2015 Homework 8 Due October, 28. 2015. Note: Except for problems that are stated explicitly, all problems are from David Logan 4th Edition. Please show all of your work (writing a list of answers is not sucient). Please indicate the people you worked with. Please staple your page together. 2 random problems will be graded (5 points each). 1. Consider a boundary value problem (this is the same equation as the #5 on HW 7) y + y = 0, 0 < 1, y(0) = 0, y(1) = 1. (a) Show that the regular perturbation method fails. (b) Use the singular perturbation method to nd the leading order uniform asymptotic approximations. 2. Use the dominant balancing method to determine the leading-order behavior of the roots of x3 + x 2 = 0, 0 < 1. 3. Find the leading order uniform approximation to y + ex y = 1, y(0) = 1, y() = 0. (Hint : Taylor series of ex ) 4. Find the leading order uniform approximation to y + ( 1 )y + y = 0, 0 < x < 1, 0 < 1+x y(0) = 0, y(1) = 1. (Hint : 1 1+x 1 x) 1 1