Consider a circular membrane r = a. For noncircularly symmetric eigenfunctions (m = 0), (9.6.18) is valid with 0 = c1(1) 0 + c2(2) 0 , where (1) 0 and (2) 0 are two mutually orthogonal eigenfunctions corresponding to the same eigenvalue 0. Here, c1 and c2 are arbitrary constants. (a) Determine a homogeneous linear system of equations for c1 and c2 derived from the fact that 1 has two homogeneous solutions (1) 0 and (2) 0 . This will be the compatibility condition for (9.6.18) with (9.6.19). (b) Solve the linear system of part (a) to determine the perturbed frequencies and the corresponding natural modes of vibration.

Limits of Infinity: In this section we are going to explore limits that go towards infinity. Limits of infinity imply, that we are going to be looking at what impacts a function if you were to let x get very large in either the positive or negative sense. Also, as we’ll soon see, these limits may also create infinity as a value. It...