(a) Obtain product solutions, = f(x)g(y), of (7.2.14) that satisfy = 0 on the four sides of a rectangle. (Hint: If necessary, see Section 7.3.) (b) Using part (a), solve the initial value problem for a vibrating rectangular membrane (fixed on all sides). (c) Using part (a), solve the initial value problem for the two-dimensional heat equation with zero temperature on all sides.Solve the initial value problem and analyze the temperature as t if the boundary conditions are * (a) u(0, y, t) = 0, u(L, y, t) = 0, u(x, 0, t) = 0, u(x, H, t)=0 (b) u x(0, y, t) = 0, u x(L, y, t) = 0, u y (x, 0, t) = 0, u y (x, H, t)=0 * (c) u x(0, y, t) = 0, u x(L, y, t) = 0, u(x, 0, t) = 0, u(x, H, t)=0 (d) u(0, y, t) = 0, u x(L, y, t) = 0, u y (x, 0, t) = 0, u y (x, H, t)=0 (e) u(0, y, t) = 0, u(L, y, t) = 0, u(x, 0, t) = 0, u y (x, H, t) + hu(x, H, t)=0 (h > 0)

L34 - 8 Suppose a particle is moving along a straight line with posi- tion function s(t), velocity v(t), and acceleration a(t). Then ▯ t2 v(t)dt = t1 ▯ t2 a(t)dt = t 1 NOTE: total distance traveled =