Question

1. Assume a dynamic system is described by the following ordinary differential equation (ODE)

1. Assume a dynamic system is described by the following ordinary differential equation (ODE): y(4) + 9y(3) + 30ij + 429 + 20y F(t) = where y = (r' y /dt'.. (a) (10 %) Let F(t) = 1 for t 0, please solve the ODE analytically. (b) (10 %) Please give a brief comment to the evolution of the system. (c) (5 %) Please give a brief comment to the stability of the system. (d) (10 %) Suppose F(t) = 1 . Please run a numerical simulation of this systern for 10 seconds 0, and y(3)(0) with initial conditions: y(0) = 0, analytical solution 0) = 0, 0) 0 to verify your

Answer 1

Given d^4 y/dt^4 + 9 d^3 y/dt^3 + 30 d^2 y/dt^2 + 42 dy/dt + 20 y = F(t) This has two parts of functions Therefore (1) complimentary function writing the auxillary equation D^4 + D^3 + D^2 + 42 D + 20 = 0 By solving the roots are -0.483 1.526 plusorminus 3.05387 i -3.5724 Therefore The (1) solution is c_1 e^-0.4803 t + c_2 e^-3.5724 + e^1.520 t [c_3 cos (3.0538 t) + c_4 sin (3.0538 t)] now (2) particular integral (D^4 + D^3 + D^2 + 42D + 20) y = F(t) = 1 y = 1/D^4 + D^3 + D^2 + 42 D + 20 times 1 1 can also be written as e^0t Therefore y = 1/D^4 + D^3 D^2 + 42D + 20 times e^0t Substituting 0 in place of D from standard solutions y = 1/20 \r\n Therefore the solution is y = c_1 e^-0.4803 t + c_2 e^-3.5724 t + e^1.526 t [c_3 cos (3.0538 t) + c_4 sin (3.0538 t)] + 1/20