小テスト解答.pdf

Se dan las siguientes ecuaciones diferenciales y sus soluciones:

(1) P(D) = D2 −4D + 3, P(D)y = x2 + x
yc = C1e3x + C2ex, y∗ = 1/3x2 + 11/9x + 38/27
y = y∗ + yc = 1/3x2 + 11/9x + 38/27 + C1e3x + C2ex

(2) P(D) = D4 −4D3 + 3D2, P(D)y = x2 + x
yc = C1 + C2x + C3e3x + C4ex, y∗ = 1/36x4 + 11/54x3 + 19/27x2
y = y∗ + yc = 1/36x4 + 11/54x3 + 19/27x2 + C1 + C2x + C3e3x + C4ex

(3) P(D) = D2 −8D + 15, P(D)y = e2xx2 + x
yc = C1e5x + C2e3x, y∗ = (1/3x2 + 11/9x + 38/27)e2x
y = y∗ + yc = (1/3x2 + 11/9x + 38/27)e2x + C1e5x + C2e3x

(4) y′′ −x −2/x y′ + 2/x y = x2ex
y = u(x)ex, u′′ + x −2/x u′ = x2
u(x) = x3/3 + C1(x2 + 2x + 2) + C2ex

(5) x2y′′ −3xy′ + y = x2 log x
P(D) = D2 −4D + 4, P(D)y = te2t
yc = (C1 + C2t)e2t, y∗ = t3/6 e2t
y = y∗ + yc = (t3/6 + C1 + C2t)e2t = [(log x)3/6 + C1 + C2 log x]x2