Question

The following equation has a polynomial input. 0.125x + 0.75x + x = y (t)--t3 + _t2 Use Simulink to plot x(t) and y(t) on the same graph. The initial conditions are zero 27 a 800 800 270 ,

Answer 1

The given differential equation is 27 270 0.125x + 0.75x x = y(t) =--t' + mt 800 800 The characteristic equation is 3 8 4 so m =-2,-4 Thus x,-ce2C2e4t For particular integral, let Then 0.125(2C 6Dt) 0.75(B 2Ct 3Dt2) 27 800 270 800 Or, (0.25C0.75BA) (0.75D 1.5C B)t 27 800 By comparison of coefficients of t of equal powers, we get 270 800 (0.25C 0.75B A) = 0 (0.75D 1.5C B) 0

27 D =-800 Solving these we get 3807 c 3200 351_ , B-6400 1323 3807 1024 Thus the 351 3807 1323 27 1024 64003200 800 And general solution is 351 3807 132327 t + C2et 1024 6400 3200 800 As x(0) 0, we have 351 C121024 And x"(0)=0 So 0 =-2c1-4c2-6400 3807 or,2 c1 4c26400 3807 351 40 6400 512 441 6400 512/ or, C2 101 233 3807 351 351 C1 1024

-9-50398Е-09 0.006677014 0.009470268 0.009047999 0.005474953 0.001519193 0.012372492 0.027588877 0.047678315 0.073123614 0.104363819 0.141787725 0.185733308 0.236490463 0.294305368 0.359385499 0.431904692 0.512007935 0.599815716 0.695427878 0.798926968 0.910381105 .029846422 0.00334125 0.01323 0.02946375 0.05184 0.08015625 0.11421 0.15379875 0.19872 0.24877125 0.30375 0.36345375 0.42768 0.49622625 0.56889 0.64546875 Chart Title 0.80956125 0.89667 0.98688375 Thus 101 -2t 233" '441 4351 3807 13232 ' 1024 6400、" 3200 1.17581625 1.27413 1157369119 1.37473875 1436731814 1.58203125 1.29298718 1.47744 800800 1.588628647 1.748698707 1.916959241 2.093424395 2.278105764 1.68831 .79607375 1.90512 2.01524625 2.12625