1. ECUACIONES
DIFERENCIAL
ES
Ejercicios
Luz Marlene Hidalgo Encarnación
30/10/2009

2. 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

3. 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

4. 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