Question

③ Resolver la ecuacion Diferencial formulase ах a x CD (O+a)

Answer 1

(D - 1)^3 (D - 2)^3 y = x^3 e^4x for Homogeneous solution Rightarrow (D - 1)^3 (D - 2)^3 y = 0 characteristic eqn Rightarrow (m - 1)^3 (m - 2)^3 = 0 Rightarrow m = 1, 1, 1, 2, 2, 2. y_p(x) = c_1 e^x + c_2 xe^x + c_3 x62 e^x + c_4 e^2x + c_5 x e^2x + c_6 x^2 e^2x find to find particular solution as (D - 1)63 (D - 2)^3 y = x^3 e^4x Rightarrow f(D) = (D - 1)^3 (D - 2)^3 y_p (x) = x^3 e^4x/f(D) = e^4x/f(D - 4) [x^3] = e^4x/(D - 4 - 1)^3 (D - 4 - 2)^3 [x^3] = e^4x/(D - 5)^3 (D - 6)^3 [x^3] = e^4x/30 [- D/5 + 1]^3 [- D/6 + 1]^3 [x^3] = e^4x/30 [1 - D/5]^3 [1 - D/6]^2 [x^3] = e^4x/30 [1 - D/5]^3 [1 - D/6]^- 3 {x^3} = e^4x/30 [1 - D/5]^3 [1 - D/6]^3 {x^3} = e^4x/30 {1 + 3 (D/5) + 12/2 (D/5)^2 + 10 (D/5)^3 + .... } {1 + 3 (D/6) + 6 (D/6)^2 + 10 (D/6)^3 + .... } {x^3}.