In applying elasticity theory to study the transverse vibrations of a beam, one encounters the equation Ely(4) (x) — yλy(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/EI; 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(0)=y'(0) = 0 and y(L) = y' (L) = 0, 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: Question: For L=1 determine the first two positive solutions to (7) cosh(rL)=sec(rL) 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.]