APPLIED MATH MASTER’S EXAM January 2007 1. Use standard techniques (i.e., not the Poincare-Lindstedt method) to find a two-term regular perturbation approximation to the solution of the ODE IVP. d2 y dy + + y = 0, t>0 2 dt dt dy y(0) = 1, (0) = 0, 0 < 1. dt 2. Give the boundary value problem which characterizes the extremal for the functional 1 Z 1 dy J(y) = 2 0 dx !2 + y(x)2 dx − 2y(1) subject to the constraints y ∈ C 2 [0, 1] and y(0) = 0. 3. Show that f () = O(g()), where → 0+ , f () = Z 2 e−x dx and g() = . 0 4. (a) Compute the eigenvalues λn and corresponding eigenfunctions φn (x) for the SLP −φ00 = λφ, φ(0) = 0, φ(π) = 0. (b) Then obtain a series representation for the solution to the boundary value problem −u00 + α2 u = f (x), 0 < x < π, u(0) = 0, u(π) = 0 Note that R1 0 sin2 (nx) dx = π/2. 5. Let D(a, b) denote the set of test functions on the interval (a, b), and let D0 (a, b) denote the set of distributions (continuous linear functionals) on the set D(a, b). (a) Does (u, φ) = [φ(0)]2 ∀ φ ∈ D(−1, 1) define a distribution in D0 (−1, 1)? (b) Let H(x) denote the Heaviside function, and let δ(x) denote the Dirac Delta function with pole at 0. Show that the δ(x) is the distributional derivative of H(x). That is, show that H 0 (x) = δ(x) in the sense of distributions.