Download Fall 2009 Math 441 Exam: Projectile Motion, Diff. Equations, Singular Perturbations - Prof and more Exams Elementary Mathematics in PDF only on Docsity! Test 1 Math 441, Fall 2009 (Due October 16th, Friday, No exception) Show all your work; use Maple to do expansions and solve equations if possible (with exceptions see below); attach your Maple calculation to the exam. 1. Consider the projectile problem with air resistance: (the page numbers are referred to the textbook) d2h dt2 = − gR 2 (R+ h)2 − k R+ h dh dt , h(0) = 0, dh dt (0) = V. (Hint: compare it with (1.32) and (1.33) on page 26) (a) Find the dimension of all quantities in the equation (including the initial values.) (hint: most are given on page 27 except k) (b) Derive equation of dimensionless quantity as in page 27, and find the number of independent dimensionless quantities of this system; three of dimensionless quantities are given in equation (1.38) page 27, now find another one which include k. (c) Find the new equation (which should be similar to (1.47)) under the change of variable (1.44) page 28, and still use the dimensionless parameter ε = V 2/(gR). (d) Find a two-term approximate solution h̄ = h̄0 + εh̄1 of the equation in (c) when ε is small. (problem 11 on page 102 is similar except without air resistance, I will show you that one Oct. 8 in class.) 2. Solve the boundary value problem: (here a > 0 is a constant) u′′ + a2u = 1, u(0) = u(π) = 0. (Do not use Maple to solve the problem, and show your work (particular solution, general solution, solving the constants, etc.)) (Hint: is there a difference if a is an integer or not? discuss when it has only one solution, infinitely many solutions, or no solution.) 3. Use regular and/or singular perturbation methods to find two-term approximate solutions of εx3 − x2 + 4x− ε = 0 when ε is small. (Hint: there are three of them, and all of them are in different scales.) 4. Assume ε is a small parameter, and consider: d2y dt2 + 9y − εy3 = 0, y(0) = 0, dy dt (0) = 1. (a) Use regular perturbation to compute the first two terms of the perturbations se- ries(find y0, y1 if y = y0 + εy1). Is your approximate solution uniformly valid? Why or why not? (b) Use Poincare-Lindstedt to find a two-term approximation of the solution. Find y0, y1 (y = y0 + εy1) and ω1 (t = τ(1 + εω1)). (c) Solve this problem numerically using Maple and compare the approximations obtained in parts (a) and (b) with a numerical solution. Discuss your findings. 5. Find the uniform approximation solution of the singular perturbation problem εy′′ + (t2 + 1)y′ − t3y = 0, y(0) = y(1) = 1 by finding the outer solution, inner solution, and matching condition.
