MATH 543: METHODS OF APPLIED MATHEMATICS I

Second Midterm Exam: 05 December 2001 Wednesday 18.00-20.00 SAZ:13

QUESTIONS: (25)1. Solve u′′ = f(x) with u(0) = α and u′(a) = β by using the Green’s function method.

(25)2. (a) Find the complete solution of the differential equation zu′′ + u′ − u = 0. First obtain the regular (analytic) and the singular solution. Solve also the initial value problem u(0) = α where α is any real constant. (25)3. Hypergeometric DE equation is defined as z(1 − z)u′′ + [c − (1 + a + b)z]u′ − abu = 0 and the solution analytic about the regular singular point z = 0 is known as the hypergeometric function F (a, b; c; z). which has a series solution

F (a, b; c; z) = \frac{Γ(c)}{Γ(a)Γ(b)} ∑_{n=0}^{∞} \frac{Γ(a + n)Γ(b + n)}{Γ(c + n)} \frac{z^{n}}{n!}

where Γ(x) is the gamma function.

(a) Confluent hypergeometric function Φ(a, c; z) defined as

Φ(a, c; z) = lim_{b→∞} F (a, b; c; z/b)

Prove that u = Φ(a, c; z) satisfies zu′′ + (c − z)u′ − au = 0

(b) Find the series (expansion) solution of Φ(a, c; z) about z = 0 (analytic solution about z = 0).

(c) Prove that

\frac{d^{n}}{dz^{n}} Φ(a, c; z) = \frac{Γ(a + n) Γ(c)}{Γ(c + n) Γ(a)} Φ(a + n, c + n; z)

(25)4. Solve u′′′′ = f(x) with u(0) = 0, u′(0) = 2u′(1), u(1) = 0, u′′(0) = 0