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APPLIED MATH MASTER’S EXAM January 2011 Instructions: Attempt 4 of the following 6 questions. Show all work. Clearly label your work and attach it to this sheet. 1. Find a two term expansion of the positive singular root X of x3 − x + = 0 2. Let y(x, ) be the solution of the boundary value problem: y 00 + y 0 + y 2 = 0 , x ∈ (0, 1) 1 1 , y(1) = y(0) = 4 2 Find the outer, inner and uniformly valid approximations of y in the limit → 0 assuming a layer exists at x = 0. 3. A functional J : A → IR where J(y) ≡ Z 1 0 1 xy(x) + y 0 (x)2 dx 2 and n A = y ∈ C 2 [0, 1] : y(0) = 1 o a) Derive the natural boundary condition for y(x). b) Find the unique extrema ȳ(x) of J over A. 4. Define the operator L and domain D by: Lu ≡ u00 D ≡ {u ∈ C 2 [0, π] : u(0) = u0 (π) = 0} a) Find all the eigenvalues λn > 0 and normalized eigenfunctions φn (x) of L b) Find the series representation g(x, ζ) of the Green’s function solving: Lu = f (x) , u(x) = Z π g(x, ζ)dζ 0 5. Find the solution of the integral equation: Z 1 ex+y u(y) dy + u(x) = e−x 0 6. Use Laplace convolution Theorem to find the bounded general solution of utt = uxx + g(t) , x>0 , u(x, 0) = ut (x, 0) = u(0, t) = 0 t>0