Question

Match the differential equation with it's particular solution form. You MUST use the method of undetermined coefficients and you MUST show all work as to how you came to your conclusions. You have ONE (1) attempt at this problem a. Aest b. Ae24 y'' - 6y' + 5y = (4t+5)e5t Vy' – 6y' + 5y = e2t ✓y'' – 6y' + 5y = est y'' – 6y' + 5y = (4t+5)e24 y'' – 4y' + 4y = 5 y'' – 4y' + 4y = e24 c. (At + B)test d. (At + B)e2 50 e. At’e2t f. Atest

Answer 1

0 y"-6y' + 5y = (4++5) est Ausciallary equation s'-65+5=0 (5-1)(3-5)=0 S= 1,5 Roots of ausciallary equation are Hence the general form of complementary function is y (t) = ce C, C₂ are Since the function et + Cest Constants. on light side is (45+5) e5t by the method of undetermined coefficients the form of the particulas solution Yp(t)=(A++B) 5+ arbitrary is But the complementary solution already contains est teem

So as to make Yolt) independent of y(t) we take the following form for particular solution Yo (+) = (At+B)test (multiplication by t) ② y"-64l+ 5y = ezt Earlier we found the general form of Complementary solution of Ye (t) = C, et + C₂ est Since the function on the light side is ezt by the method of Undetermined coefficients the form of the posticular solution is Ip (t) = A p2t (no common telms with y(t)) (3) y"-691 + 5y = est complementary solution y(t) = C, et + C₂ e 5 By the method of undetermined coefficients, the form of particular 5t solution is 464) = Ae5+ But y(t) contains est teem. So as to make up (t) independent of Ye (t) we choose the following form YpCt) = Ate 5+ (multiplication with t) 4 y"-6y + 5y = (4t+5) ezt complementary solution y(t) = C, et + Cest By the method of undetermined Golticente the form of particulas Ip (t) = (At+B) ezt (no Common terms with y(t)) solution is

55 Yp(t) = ne y – 4y + 4y = 85t ③ y' 44'+4y= est Ausciallary equation is 5-45+4=0 (S-2) - 0 s=2,2 Since the roots are repeated the general form of ausciallary equation is y(t) = C, e 2t + Cat ezt By the method of undetermined Coefficients, the form of particular solution is (no Common term with y(t)) ③ y' 44'+ 4y= ezt But ų (t) y"-byl- y(t) = C, e 25+ Cat ezt Complementary solution is By the method of undetermined Coefficients, the form of particular solution is But ye(t) contains ezt teem, so we choose up (t) = At ezt Contains also tet teem, so we choose Yo (t)= At ezt Jp = Aezte (At+B) test A pet y"-Gy'+5y = (4t+5) e5t y" - byl + 5y = ezt y" by'+ 5y = est Atest (A6+B) et + 5y = (46+5) = 26 Aest Atreet y"_4y +44