**TEST 1 MA-4451**

**1.** Formulate the problem of the transverse vibration of a string of length L, on which the gravitational force acts at
each point. The endpoints of the string are fixed. Initially
the position of the string is given by the function *f*(*x*)
and the string is at rest.(20 pts.)

**2.** **a.** Find the sine Fourier series for the
function:(15 pts.)

**b.** What is the value of the expansion for *x* = 0 ?
(5 pts.)

**c.** Find the cosine Fourier series for the
function:(15 pts.)

**d.** What is the value of the expansion for *x* = 0 ?
Justify the answer.(5 pts.)

**3.** Show that the polinomials:

are orthogonal on the interval [-1,1]. Are the two functions normalized ? (20 pts.)

**4.** Determine the type (elliptic, parabolic or hyperbolic)
of the following partial differential equations:(20 pts.)

Wed Sep 11 08:20:47 EDT 1996