**MATH3271**

**FW03**

**Assignment
4 Due date: Friday, Nov. 14**

1. (a) Solve the problem of a
vibrating circular membrane of radius 1 if it is initially
undisplaced and has an initial velocity given by g(r,j)
= (1- r^{2})sin(2
j). Take c = 1.

(b) Modify your MAPLE program from assignment 3 to calculate the first 10 coefficients of the series solution and plot the tenth partial sum of this series for several values of t.

2. (a) Find the steady-state temperature of a disk of radius 1 if the temperature on the boundary is given by 1 + sin(2 j).

(b) Use MAPLE to plot this solution.

3. (a) From the proof of the orthogonality integral (see section 4.8 of the text) deduce that

if m ¹
n and l_{m} = a_{m}/a
where a_{m }is a zero of J_{0}(x)
OR a zero of J_{0}¢(x)
with a similaar definition for l_{n}.

(b) Find the steady-state
temperature of a cylinder of radius a and height h if the side is
__insulated__ and the temperature on the bottom is kept a zero
degrees while the temperature distribution on the top is given by
f(r).

4.(a) Find a solution to the Poisson's equation

Ñ_{2}^{2}
u(r,j) = r
cos(j), 0 < r
< 1, 0 < j < 2p

with boundary condition u(1,j) = 0.

(b) Modify your MAPLE program from question 1 to calculate the first 10 coefficients in the series solution and plot the tenth partial sum of this solution.

5. For each of the following differential equations, determine if x = 0 is an ordinary point, a regular singular point or neither. If it is a regular singular point, solve the indicial equation and state which case of the Frobenius method applies.

(i) 4x^{2}y¢¢
- 14xy¢ - (18 - x)y = 0 (ii) xy¢¢
+ y¢ - (1+x)y/x = 0

(iii) y¢¢
+ (1 - x^{2})y¢ + xy =
0 (iv) x(1-x)y¢¢ + (1-3x)y¢
- y = 0

(v) 4xy¢¢
+ 2xy¢ + y = 0 (vi) x^{3}y¢¢
+ x^{2}y¢ - y = 0