Use separation of variables to solve the following PDE: 22 =-2044 774 (1) This PDE describes lateral deflection of a beam, similar to the string equation. The constant 2 = El/pA. E is the Young's modulus of the material, I is the second area moment of the beam cross-section, p is the density of the material, and A is the cross-sectional area. Some points that should help your solution: Define the constant of separation as 34. This will simplify the solution of the Sturm-Liouville problem With this constant, the solution to the spatial equation is F(x) = Acos(x) Bsin(Bx) C cosh(Bx) Dsinh(Bx) (2) With this constant, the solution to the time equation is G (t) = Q cos(cB?t) Rsin(cB?t) (3) 1. Separate Equation (1) and solve the resulting ODEs to get Equations (2) and (3). 2. If the beam is simply-supported with initial deflection f(x) and no initial velocity, solve for the deflection U(x, t). As a reminder, a simply supported beam has zero deflection and moments at the end-points. According to Euler-Bernoulli beam theory, this gives the following boundary condtions: u(0,t) = U(L, t) = Uxa (0, t) = Uxz (L, t) = 0 (4)