C Transverse Vibrations of a Beam In applying elasticity theory to study the transverse vibrations of a beam, one encounters the equation Ely (4)(x)-yay(x) = 0, where y(x) is related to the displacement of the beam at position x; the constant E is Young's modulus; I is the area moment of inertia, which we assume is constant; y is the constant mass per unit length of the beam; and A is a positive parameter to be determined. We can simplify the equa- tion by letting := yλ/El; that is, we consider (5) y(4)(x) - ry(x) = 0. When the beam is clamped at each end, we seek a solution to (5) that satisfies the boundary conditions (6) y(L) =y' (L) = 0, y(0)=y'(0) = 0 and where L is the length of the beam. The problem is to determine those nonnegative values of r for which equation (5) has a nontrivial solution (y(x) = 0) that satisfies (6). To do this, proceed as follows: (a) Show that there are no nontrivial solutions to the boundary value problem (5)-(6) when r = 0. (b) Represent the general solution to (5) for r > 0 in terms of sines, cosines, hyperbolic sines, and hyperbolic cosines. (c) Substitute the general solution obtained in part (b) into the equations (6) to obtain four linear algebraic equations for the four coefficients appearing in the general solution. (d) Show that the system of equations in part (c) has nontrivial solutions only for those val- ues of r satisfying (7) cosh (rL) = sec(rL). (e) On the same coordinate system, sketch the graphs of cosh (rL) and sec (rL) versus r for L = 1 and argue that equation (7) has an infinite number of positive solutions. (f) For L = 1, determine the first two positive solutions to (7) numerically, and plot the corresponding solutions to the boundary value problem (5)-(6). [Hint: You may want to use Newton's method in Appendix B.]