By the principles used in modeling the string it can be shown that small free vertical vibrations of a uniform elastic beam are modeled by the fourth-order PDE ди at² Ox4 ,where c²=El/pA (E = Young's modulus of elasticity,I = moment of intertia of the cross section with respect to the y-axis in the figure, A = cross-sectional area). Substituting u=F(x)G(t) into the equation, show that F4/F = -G/c² G = ẞ4 = const, F(x) A cos ẞx + B sin ẞx + C cosh Bx + D sinh Bx, G(t) = acos cẞ² + b sin cẞ² t. x=L