Question

Exercise 3 (6 marks) Consider the forced mass-dampener-spring system that is represented by the differential equa- tion, mx" (t) + ca' (t) + k2(t) = e-t-e-2t where 1. Solve this IVP by using the Method of Undetermined Coefficients (MUC). 2. Solve this IVP by using the Variation of Parameters Method (VOP).

Answer 1

Put 끼 = 1.c=5.k=6,x(0) = 1/2,x'(0) = 112 nmx''t cr't kx = e-t-e-2t (1) Undetermined coefficients Chararcteristic equation of the homogeneous differential equation x"5x'6x 0 is solution is xt = cic" + c2 e- A guess for particular solution s xAe Be, 2r Substitute x^.x', andx", in the original differential equation 2Ae A but B can not be evahuated, so this method doesn't work to find the solution of the given differential equation. But the solultion surely can be found using variation of parameters (2) variation of parameters 2r 2r Particular solution is dt 3r e

General solution : x(t)-+ x,-qc" + c2e® +-e" + e-2t-te-2t 2 Use x(0)= 1/2.x'(0)=1/2 to find c1 and c2 2 2 Hence, x(at ) =-2e-st