mate2.3 Ecuaciones lineales homogéneas con coeficientes constantes
1. 3y’’ – y’=0

m(3m-1)=0
m=0;m=1/3


2. 2y’’ + 5y’=0

m(2m+5)=0
m=0;m=-5/2

3. y’’ + 16y=0


;
Raíces: +4 ; -4


4. y’’ + 8y=0
8=0




±2


5. y’’+9y=0









6. 4y’’ + y=0
4










7. y’’ – 3y’ + 2y=0



4 ;

8. y’’ – y’ – 6y=0


; m=-1

9. y’’ + 8y’ +16y=0





10. y’’ -10y’+25y=0

(m-5) (m-5)


11. y’’ – 3y’ -5y=0













12. y’’ -4y’-y=0











13. 12y’’ -5y’ -2y=0

















14. 8y’’+2y’-y=0











15. y’ -4y’ + 5y=0



















16. 2y’’ + 3y’ +4y=0







17. 3y’’ + 2y’ +y=0





















18. 2y’’ + 2y’ +y=0









19. y’’+16y=0 y(0)=2; y’(0)=-2










y(0)=2 y’(0)=-2
-2=4



20. y’’ –y=0 y(0)=y’(0)=1

(m-1)(m+1)


0= +



21. y’’ +6y’ +5y=0 y(0)=0; y’(0)=3



y’(0)=3 ;


(sustituimos C1 en Yc)

22. y’’-8y’+17y=0 y(0)=4; y’(0)=-1









Y(0)=4; y=

Y’(0)=-1----------4


23. 2y’’ -2y’ +y=0 y(0)=-1; y’(0)=0







Y(0)=-1 ;

Y’(0)=0;
[sustituimos el valor de c1]



24. y’’ – 2y’ +y=0 y(0)=5, y’(0)=10














25. y’’+y’+2y=0 y(0)=y’(0)=0










26. 4y’’-4y’-3y=0 y(0)=1; y’(0)=5






















2.5Metodos de coeficientes indeterminados
1. y’’- 9y = 54
m2- 9=0 (0)-9(A)=54(m+3)(m-3)=0 A=-54/9
R1=-3; R2=3 A=-6
Yc=c1e-3x+c2e3x y=c1e-3x+c2e3x-6
Yp=A SOLUCION GENERAL.
Y’p=0
Y’’P=0

2. 2y’’- 7y’ + 5y = -29
x=-b±√b2-4ac /2a Yp=A 2(0)-7(0)+5A=-29
x=7±√(7)2-4(2)(5) /4Y’p=0 5A=-29
x=7±√49-40 /4 Y’’P=0 A=-29/5
x=7±3 /4
X=10/4 x=4/4 Y=c1e10/4x+c2ex-29/5
R1=10/4; R2=1 SOLUCION GENERAL.
Yc=c1e10/4x+c2ex

3. y’’ + y’ = 3
m2+m=0 0+A=3
m (m+1)=0 A=3
m=0 m=-1
R1=0; R2=-1 Y=c1+c2e-x+3x
Yc=c1e0+c2e-x SOLUCION GENERAL.
Yc=c1+c2e-x
Yp=Ax
Y’p=A
Y’’P=0




4. y’’’ + 2y’’ +y’ = 10
m3+m2+m=0 0+2(0)+A=10
m(m2+m+1)=0 A=10
m (m+1) 2=0
m=0 m=-1
R1=0; R2=-1 Y=c1+c2e-x+c3xe-x+10x
Yc=c1e0+c2e-x+c3xe-x SOLUCION GENERAL.
Yc=c1+c2e-x+c3xe-x
Yp=Ax
Y’p=A
Y’’P=0
Y’’’P=0

5. y’’ + 4y’ +4y = 2x + 6
m2+4m+4=0 A+4Ax+4(Ax+B)=2x+6
(m+2)2=0A+4Ax+4Ax+4B=2x+6
m=2 m=2 A+8Ax+4B=2x+6
R1=2; R2=2 8Ax=2x 4B=6
Yc=c1e2x+c2xe2x 4A=0 B=6/4
Yp=Ax+B A=0
Y’p=Ax
Y’’p=A Y=c1e2x+c2xe2x+6/4
SOLUCION GENERAL.
6. y’’ + 3y’ = 4x – 5
m2+3m=0 y’’p=2A 2A+3(2Ax+B)=4x-5
m(m+3)=0 2A+6Ax+3B =4x-5
m=0 m=-3 6Ax=4x 3B=-5
R1=0; R2=-3 6A=4 B=-5/3
Yc=c1e0+c2e-3x A=4/6
Yc=c1+c2e-3x
Yp=Ax2+Bx Y=c1+c2e-3x+4/6x2-5/3x
y’p=2Ax+B SOLUCION GENERAL.

7. y’’’ + y’’ =8x2
m3+m2=0 24Ax+6B+12Ax2+6Bx+2F=8x2
m2 (m+1)=0 12 Ax2=8x2 24Ax+6Bx=0x 6B+2F=0
R1=0; R2=0; R3=-112A=8 24(8/12)+6B=0 6(-16/6)+2f=0
Yc=c1e0+c2xe0+c3e-x A=8/12 16+6B=0 -16+2F=0
Yc=c1+c2x+c3e-x6B=-16 2F=16
yp=Ax4+Bx3+Fx2 B=-16/6 F=8
y’p=4Ax3+3Bx2+2Fx
y’’p=12Ax2+6Bx+2F Y=c1+c2x+c3e-x+8/12x4-16/6x3+8x2 y’’’p=24x+6B SOLUCION GENERAL.

8. y’’ – 2y’ + y = x3 +4x
m2+2m+1=0 6Ax+2B-2(3Ax2+2Bx+F)+Ax3+Bx2+Fx+R=x3+4x
(m+1) 2=0 6Ax+2B-6Ax2-4Bx-2F+Ax3+Bx2+Fx+R=x3+4x
m=-1 m=-1 Ax3=x3 -6Ax2+Bx2=0x2 6Ax+Fx=4x 2B-2F+R=0
R1=-1; R2=-1 A=1 -6(1)+B=0 6(1)+F=4 2(6)-2(4/6)+R=0
Yc=c1e-x+c2xe-x B=6 F=4/6 6-8/6+R=0
Yp =Ax3+Bx2+Fx+R 14/3+R=0 y’p=3Ax2+2Bx+F R=-14/3
y’’p=6Ax+2B Y=c1e-x+c2xe-x+x3+6x2+4/6x-14/3
SOLUCION GENERAL.
9. y’’ – y’ -12y = e4x
16Axe4x +4Ae4x-4Axe4x+Ae4x-12(Axe4x)=e4x
(m-4)(m+3) 16Axe4x +4Ae4x-4Axe4x+Ae4x-12Axe4x =e4x
m=4 m=-3 4Ae4x +Ae4x =e4x
R1=4; R2=-3 5Ae4x =e4x
Yc=c1e4x+c2e-3x 5A=1
Yp=Axe4x A=1/5
Y’p=(Ax)(4e4x)+(e4x)(A)
Y’p=4Axe4x+Ae4x Y=c1e4x+c2e-3x+1/5xe4x
Y’’p=(4Ax)(4e4x)+4Ae4x SOLUCION GENERAL.
Y’’P=16Axe4x +4Ae4x


10. y’’ – 2y’ + 2y = 5e6x
x=-b±√b2-4ac /2a 36Ae6x-2(6Ae6x)+2(Ae6x)= 5e6x
x=-2±√(2)2-4(1)(2) /2 36Ae6x-12Ae6x+2Ae6x= 5e6x
x=-2±√4-8 /2 24Ae6x+2Ae6x= 5e6xx=-2±√4 /226Ae6x= 5e6x
X=-2+2/2 x=-2-2/226A=5
R1=0; R2=-2 A=5/26
Yc=c1e0+c2e-2x
Yc=c1+c2e-2x Y=c1+c2e-2x+5/26e6x
Yp=Ae6x SOLUCION GENERAL.
Y’p=6Ae6x
Y’’p=36Ae6x
11. yŽŽ-2y’-3y= 4ex-9
yŽŽ-2yŽ-3y=0yp=4ex
m2-2m-3=0y’p=4ex
(m-3)(m+1)=0yŽŽp=4ex
m1=3 m 2=-1 4ex-2(4ex)-3(4ex)=4ex
yc=C1e3x+C2e-x4ex-8ex-12ex=4ex
-16ex=4ex 4ex/-16ex yp=4/-16
-16 a1=-9 a1=-9/-16 yp=4/-16 +-9/-16
Yg= C1e3x+C2e-x+4/-16-9/-16

12. yŽŽ+6yŽ+8= 3e-2x+2x
y’’+6y’+8y=0yp=ke-2xxyp=a1x+a0
m2+6m+8=0y’p=-2ke-2x+ke-2xy’p=a1
(m+4)(m+2)=0y”p=4ke-2x-2ke-2xy”p=0
m 1=-4 m 2=-24ke-2xx+6(-2ke-2xx+ke-2x)+8ke-2xx=3e-2x
yc= C1e-4x+C2e-2x4ke-2xx+12ke-2xx+6ke-2x+8ke-2xx= 3e-2x
6ke-2x=3e-2x
K=3e-2x/6e-2x k=3/6
6 a1+8 a1+8 a0=2x8 a1x=2x a1=2x/8x a1=2/8
Yg= C1e-4x+C2e-2x+2/8x-3/16

13. yŽŽ+25y= 6senx
y’’+25y=0
m2+25=0
(D2+25)y=0
yc= C1sen25x+C2cos25x
yp=a1senx+a2cosx
y’p=a1cosx-a2senx
y”p=-a1senx-a2cosx
-a1senx-a2cosx+25(a1senx+a2cosx)=6senx
-a1sex-a2cos+25 a2senx+25 a2cosx=6senx
(-a1+25 a)(senx+(-a2+25 a2)cosx=6senx
24 a1senx=6senx
a1=6senx/24senx a1=6/24 24 a2=0 a2=0
yp=6/24senx+0(cosx)
Yg= C1sen25x+C2cos25x+6/24senx

14. y’’+4y= 4cosx+3senx-8
y’’+4y=0yp=acosx+bsenx-8
m2+4y’p=-aenx+bcosx
(m+2)2yŽŽp=acosx+bsenx
m 1=-2 m2=-2
yc= C1e-2x+C2xe-2x acosx+bsenx+4(acosx+bsenx)=4cosx+3senx
acosx+bsenx+4acosx+4bsenx= 4cosx+3senx5acosx+5bsenx=4cosx+3bsenx
a=4cosx/5cosx a=4/5 cosx b=5senx/3senx b=5/3senx
Yg= C1e-2x+C2xe-2x +4/5cosx+5/3senx




15. yŽŽ+6y’+9y= -xe4x
y’’+6y+9=0yp=-axeb4x
m2+6m+9=0y’p=-a4xeb4x
(m+3)(m+3)=0y’Žp=-a16xeb4x
m1=-3 m 2=-3axeax+beax
yc=C1e-3x+C2xe-3-a16xe4x +6(-a16xe4x )+9(-axe4x )
-a16xe4x -96xe4x -9axe4x =-xe4x
a1=25xe/-xe4xa1=25xe4xa2=-96xe4x/-xe4x a2=-96xe4x
Yg=C1e-3x+C2xe-3 +25xe4x-96xe4x
16. y’’+3y’-10y =x(ex+1)
y’’+3y-10=0
m2+3m-10=0
(m+5)(m-2)=0
m1=-5 m 2=2
yc= C1e-5x+C2e2x
17.yŽŽ-y=x2ex+5
y’’-y=0
m2-1=0
(m-1)(m+1)
m1=1 m 2=-1
yc=C1ex+C2e-x
18. y’’+y’+y = x2e-x
m2+2m+1=0
(m+1)(m+1)=0
m1 =-1 m 2=-1
yc= C1e-x+C2xe-x5
19. y’’ -2y’+5y=





2±




















20. y’’+y’+ 1/4y=

















21.
















22.




















23.


25. y’’’-3y’’+3y’-y=





























26. y’’’-y’’+y’-y=x




























27. y’+25y=20sen5x










28. y’’+y=4cosx-senx





















29.y’’ +y’+y=xsenx










(B+C+2D)cosx+Dxcosx+(-A-2B+D)senx-Bxsenx-xsenx










30. y’’+4y=




























sen2x+

31.y’’ –y’=























32.2y’’’-3y’’-3y’+2y=(




























33.-Y”-64y=16 y(0)=1 y’(0)=0
M^2-64=0
(m-8) (m+8)m=8 m=-8
Y=c1e^8x+C2e^-8x
1=C1e^8(0)+C2e^-8(0)
*C1+C2=1
Y’=8C1e^8x-8C2e^-8x
1=8C1e^8(0)-8C2e^-8(0)
*8C1-8C2=0
C1=C2
C2+C2=1
2C2=1
C2=1/2
C1=1/2
Yp=1/2e^8x+1/2e^-8x











34.-y”+y’=x y(0)=1 y’(0)=0
m^2+m=0
m(m+1) m=0 m=-1
y=c1+c2e^-x
1= c1+c2e^-(0)
1=c1+c2
Y’=-c2e^-x
0=-C2e^(0)
C2=0 C1=1
Yp=1
35-y”-5y’= x-2y(0)=0 Y’(0)=2
M^2-5m=0
M(m-5) m=0, m=5
Y=c1+c2e^5x
0= c1+c2e^5(0)
*C1+c2=0
Y’=5C2e^5x
2=5C2e^5(0)
*5C2=2
*C2=2/5 *C1= -2/5 Yp=-2/5+2/5e^5x

36.-y’’+5y’-6y=10e^2x : y(0)=1, Y’(0)=1
M^2+5m-6=0
(m+6)(m-1)
M=-6 M=1
Y=c1e^-6x+C2e^x
1=c1e^-6(0)+C2e^(0)
*C1+c2=1
y’=-6c1e^-6x+C2e^x
1=-6c1e^-6(0)+C2e^(0)
-6c1+c2=1
C2=1+6c1
C2=1 C1=0
Yp=0e^-6x+e^x
Yp=e^x






37.-y’’+y=8cos2x-4senx ; y(π/2)=-1 y’(π/2)=0
M^2+1=0
M^2=-1
M=
M=i
Y=(c1+cos x+c2sen x)
-1=(c1+cos(π/2)=+c2sen (π/2))
0 1
-1=c2 C2=-1
Y’=-c1sen x+c2 cos x
0=-c1sen 90+c2 cos 90
0=-c1
-c1=0

Yp=-sen x







38.- y’’’+2y’’+y’=xe^x+5 y(0)= 2 y’(0)=2
M^3+2m^2+m=0
M(m^2+2m+1)
M=0 M=-1 M=-1
Y=c1+c2e^-x+c3xe^-x
2= c1+c2e^-(0)+c3xe^-(0)
C1+c2=2
Y’=-c2e^-x+c3e^-x- c3e^-x
2=-c2e^-(0)+c3e^-(0)- c3e^-(0)
2=-c2+c3
Y’’=c2e^-x-c3e^-x+ c3e^-x- c3e^-x
-1= c2e^-(0)-c3e^-(0)+ c3e^(0)- c3e^-(0)
-1=c2-2c3

*2=-c2+2c3 2=-c2-1 c1+c2=2
*-1=c2-2c3 -c2=3 c1-3=2
-1=c2-2c3 c2=-3 C1=5
1=-c3---- C3=-1

Yp=5-3e^-x-1xe^-x


39.




X=2+2i




Sust
Y’’-4y’+8y=x3