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Ecuaciones Diferenciales[1]

The document contains solutions to three differential equations: 1) y'' - 3y' = 8e^3x + 4sinx with solution Y = (C1 + C2e^3x) + (8/3)Xe^3x + (6/5)cosx - (2/5)sinx 2) y'' + y = xcosx - cosx with solution Y = C1cosx + C2sinx + (1/4)xcosx - (1/2)xsinx + (1/4)x^2sinx 3) y'' + 4y = 4cosx + 3sinx - 8 with solution Y

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ECUACIONES DIFERENCIAL ES Ejercicios Luz Marlene Hidalgo Encarnación 30/10/2009
ECUACIONES DIFERENCIALES 1. y’’ – 3y’ = 8 e^3x + 4 sen x Solución: m^2 – 3m = 8 e^3x + 4 sen x m (m-3) = 0 ; m1 = 0 , m2 = 3 Yc = C1 + C2 e^3x (D-3) e^3x = 0 (D^2 + 1) sen x = 0 (D – 3) (D^2 + 1) (D – 3) (D^2 + 1) (D^2 – 3D) y = 0 Ecuación auxiliar: (m-3)(m^2 + 1)(m^2 – 3m) = 0 = m (m-3) ^2 (m^2 + 1) = 0 Y = (C1 + C2 e^3x) / Yc + C3 X e^3x+ C4 y cos x + C5 sen x Yp = A x e^3x + B cos x + C sen x Simplificando: Y’’p – 3y’p = 3AXe^3 + (-B – 3C)cos x + (3B – C) sen x = 8e^3x + 4 sen x 3A = 8 -B – 3 = 0 5B – C = 4 A = 8/3 B = 6/5 C = -2/5 Yp = 8/3 X e + 6/5 cos x – 2/5 sen x Y = C1 + C2 e^3x + 8/3 X e^3x + 6/5 cos x – 2/5 sen x Luz Marlene Hidalgo Encarnación. 9110112
ECUACIONES DIFERENCIALES 2. y’’ + y = x cos x – cos x (m^2 – 1) = 0 m = ± √1 y = C1 cos x + C2 sen x (D^2 + 1) ^2 (D^2 + 1) ^2 (D^2 + 1) y = 0 (D^2 + 1) ^3 y =0 y= C1 cos x + C2 sen x + C3 x cos x + C4 X sen x + C5 x^2 cos x + cos x^2 sen x Sustituyendo: Yp = A x cos x + B x sen x + C x^2 cos x + E x^2 sen x Simplificando: Y’’p + Yp = 4 E X cos x – 4 C X sen x + (2B + 2C) cos x + (-2A + 2E) senx = x cos x – cos x 4E = 1 -4C = 0 2B+2C = -1 -2ª +2E = 0 E=¼ C=0 B=-½ A=¼ Y= C1 cos x + C2 sen x + ¼ x cos x – ½ x sen x + ¼ x^2 sen x Luz Marlene Hidalgo Encarnación. 9110112
ECUACIONES DIFERENCIALES 3. Y ‘’ + 4 y = 4 cos x + 3 sen x – 8 m^2 + 4m=0 m^2 +4=0 m^2 = -4 m = ± √-4 m1 = m2= 2i Y c = C1 cos 2x + C2 sen 2x Anulador 4 cos x + 3 sen x - 8 (m^2 + 1) (m^2 + 2) = 0 m (m^2 + 1)” = 0 m3 = 0 m4, m5, m6, m7 = - i Y’ p = C3 + C4 cos x + C5 sen x + C6 x cos x+ C7 sen x Y” p=-C4cosx - C5senx - C6xcosx - C6senx - C6senx - C7xsenx + C7cosx + C7cosx SIMPLIFICANDO Y” p = - C4cosx – C5senx – C6xcosx – 2 C6 sen x – C7xsenx + 2C7cosx SUSTITUYENDO -C4cosx - C5senx – C6xcosx - 2C6senx - C7 x sen x + 2C7cosx + 4C3 + 4C4cosx + 4C5senx + 4C6xcosx + 4C7xsenx = 4cosx + 3senx – 8 3C4cosx + 3C5senx + 3C6xcosx -2C6senx + 3C7xsenx + 2C7cosx + 4C3= 4cosx + 3senx - 8 4C3=-8 C3=-2 3C4+2C7=4 C4=4/3 3C5-2C6=3 C5=1 3C6=0 C5=0 3C7=0 C6=0 C7=0 Y = C1cos2x + C2sen2x– 2 + 4/3 cosx + senx Luz Marlene Hidalgo Encarnación. 9110112