n=0 c n x n+r , determine the indicial equation. (5 %) Please also determine the indicial equation of −x 2 y 00 (x) − 2xy 0 (x) + 56y(x) = 0. (5 %)

Å Cɽ¸×ç´½ ˙çÍ ₂₀₀₃ ˙bç ₍ ø ₎ ‚5 (closed book)

1 Given an ODE

4x y ⁰⁰ (x) + 2 y ⁰ (x) + y(x) = 0

Is x = 0 singular point ? (5 %) Is x = 0 regularly singular point ? (5 %) By changing the variable of x = t ² , transform the original ODE to the new ODE for Y (t) and solve Y (t) and y(x), where y(x) = Y (t). (10 %) By setting y = P ∞

n=0 c _n x ^n+r , determine the indicial equation. (5 %) Please also determine the indicial equation of −x ² y ⁰⁰ (x) − 2xy ⁰ (x) + 56y(x) = 0. (5 %)

2 Fill in the table [A], [B], [C] ... [J] (20 %)

Sturm-Liouvilli System (py ⁰ ) ⁰ + qy = −λρy

Function ODE p(x) q(x) ρ(x) λ

Legendre polynomial (1 − x ² )y ⁰⁰ − 2xy ⁰ + N (N + 1)y = 0 [A] 0 1 [B]

Bessel function x ² y ⁰⁰ + xy ⁰ + (x ² − ν ² )y = 0 [C] x [D] −ν ²

Simple Harmonic Motion function [E] 1 0 1 ω ²

Hermite function y ⁰⁰ − 2xy ⁰ + 2αy = 0 [F ] 0 e ^−x

2α Laguerre function xy ⁰⁰ + (1 − x)y ⁰ + ny = 0 [G] 0 [H] n Chebyshev polynomial (1 − x ² )y ⁰⁰ − xy ⁰ + n ² y = 0 [I] 0 [J ] n ² Write down the orthogonal properties of Legendre and Chebyshev polynomials. (10 %) 3 Find the Laplace transform of 1, t, cos(t), sin(t), e ^t , e ^−t , cosh(t), sinh(t). (20 %) 4 We define the following functions,

u(t) = 1, t > 0; otherwise u(t) = 0 p(t) = 1, t > 1; otherwise p(t) = 0 q(t) = 1, 0 < t < 1; otherwise q(t) = 0

Please find the Laplace transform of u(t), p(t) and q(t). (15 %) What is the definition of convolution ? Write down the mathematical formula and its geometric meaning. (10 %) Please find the convolution of q(t) and q(t). (10 %)

5 If we define the correlation function of a(t) and b(t) by c(t) =

Z ∞

−∞

a(τ )b(t + τ )dτ

Please explain its geometric meaning (5 %) and determine the correlation of q(t) and q(t). ( 10%)

½×´ Í 2003 ‚5 (closed book)

æf:f inal03.ctx f: 05:50 Jan./15/2003