Lecture 24:

8. (Exercise Lecture 25) Consider the undamped massspring oscillator with equation of motion my\" + kg = sin (wt) subject to y(0) = y'(0) = 0. Find the solution if m = 2, k = 32, and w = 4. Sketch the graph of the solution for 0 S t g 10. Hint: The Addendum to the Method of Undetermined Coefcients must be used to solve this problem. Addendum to Method of Undetermined Coefficients Given a nonhomogeneous DE y" + py'+ q y= g(t)|where git) is a solution of the corresponding homogeneous DE , the trial solution must be multiplied by t until it no longer duplicates a solution of the homogeneous DE . Example: Find a particular solution of y"+ 4y'+ 34 = est Solution: The homogeneous DE has characteristic equation r + 4r+3 =0 sor=-3,-1. The complementary solution is thus yelti= Get + Cze t Ordinarily, the trial solution would be ypt)= Aest, but this duplicates part of the complementary solution. So, the modified trial solution should be yp(H = Ate 3t yp (t) = Aest -3Ate3t - Up"(ti= -6Ae *+9Ate 3t Substituting into the nonhomogeneous DE leads to (- 6 Ale 3+ 9Ate 3t) + 4( Ae- 3t- 3 At , 3t ) + 3 A test= est Equating like terms: -6A +4A -2A = 1 test; 9A -12A +3A = 0 0 =0 Hence Yo(t) = tte 3t Example: Solve the initial value problem : y + 2y' + 10y = cosltl subject to ylo)= 0, y'(0)=0. Solution: @ complementary: r?+2r+10=0-(r+1) +9=0 -> r=-1+3i. yelthere cos ( 3t ) + Czet sin (3+) 2 particular: 9Lt) = cost YPLH = Arcos (t) + Azsin(t) ( This is okay ble it does not duplicate yelth . ) Yp(t ) = - Asin(t) +Azlos(t) Yo"(LY = - Arcos(t - Azsin(t) Substitute : ( - Alcost - Azsint ) + 2 (- Aisint + Az cost) + 10 (A, cost + Azsint) = cost Equate like terms: cost : -A1 +2A2 +LOA, = 1 2 [9A, + 2A2=1] AL= 85 Sint: -A2-2A +10A2 0 9[-2A1 + 9A2=0] Al= ZAZ = 85 85A2 = 2 Hence Yolt)= 85 cost + 85 sint. (3 general solution y(t) = ypit) +yh(t) = as cost+ 85 sint + Ge cos(3+) + Getsin(3t) Hence Up (t)= -9 85 sint + cost - Ge"cos(3t) - 3 Ge sin(3t) - Cze sin(3+) + 3Ce- cos(3t). (2 4 initial condition ylol =0 = 85 +C, y' (0 ) = 0 = 85 - 6 +362 255 The solution of the IVP is thus 85 cost + 2 sint 85 e cos ( 3 t ) - IL 255 e sin ( 3t )