1. Given the following initial value problem: 6y =0, y(0) = 5, , y...

1. Given the following initial value problem: 6y =0, y(0) = 5, , y'(0) = -9. (1) Consider a series solution of (1) of the form 00 y(x) = [ an xn . n=0 (a) (8 points) Determine a recurrence relation for the coefficients an. (b) (9 points) Determine the series solution to the initial value problem (1) . How many non-zero terms are in the solution? 2. (16 points) Determine the Fourier series for the function f on the interval [ - IT, T ] when f is given by 0, -T f(x) = , f(x + 2TT) = f(x). x, 03. Consider the Sturm-Liouville boundary value problem: X"(xx = - AX(x), , 0 x x < TT, X (0) = 0, X(T) + X'(IT) = 0. (a) (5 points) Determine if l = 0 is an eigenvalue of (2). (b) (5 points) Show that (2) has no negative eigenvalues. (c) (6 points) Determine a function f(1) such that the roots of the equation = are the positive eigenvalues of (2). (d) (2 points) Based on your solution from (a)-(c), state the eigenfunctions for (2).4. (16 points) Consider the semi-infinite problem for wave propagation: d2y = 4 4.22 y x > 0, t > 0, Ot² Ox2 , y(x, 0) = f(x), x>0, by (x, 0) = 0, x > 0, ot y(0,t) = 0, t>0, where the initial displacement profile f(x) is the function shown below: f(x) 1 x 1 2 3 4 5 6 7 Graph the solution y (x,t) at the times t = = t = 1, t= and t = 2. Note: two pages of graph paper are included at the end of this exam. Questions 1-2 are on the previous page, and Questions 5-6 are on the next page * 5. Consider the initial value problem ou + 2 t Ou = 0, -00 < x < 00, t > 0 Ot Ox u(x,0) = 14xx, -00 < x < 80. (a) (8 points) Sketch at least three of the characteristics on the characteristic xt -plane. - (b) (8 points) Determine the solution u(x,t) of (3).6. Consider the following initial boundary value problem for vibrations of a circular membrane with wave speed C: d2y = c2 Ot2 a2y Or2 + - r 1 ou ) , 0 0, (i) Or y(b,t) = 0, t > 0 , (ii) y(r,0) = f(r), 0(iii) by = 0 < r < b. ot (iv) Assume that the solution y(r,t) is bounded at r = 0. (a) (6 points) Assume that y(r,t) = R(r) T '(t) Derive the ordinary differential equations that are satisfied by R(r) and T(t). (Note: you can leave these equations in a form involving the separation constant 1.) (b) (2 points) State the homogeneous boundary condition for R(r). (c) (5 points) Use the differential equation from (a) and the boundary condition from (b) to determine if there is a bounded non-trivial solution R(r) when > =0. (d) (2 points) Show that the equation derived for R(r) in (a) can be expressed in the form = 0 (4) for some constant a # 0. (e) (2 points) Given that the general solution of (4) is R(r) = C1 Jo(ar) + C2 Yo(ar) (5) where Jo(r) is bounded as r 0 and yo(r) - -00 as r 0, briefly explain the physical motivation for setting C2 = 0 in (5) when determining R(r). .