Trabalho de Cálculo III - EDH, Método dos coeficientes Indeterminados

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Trabalho de Calculo III
Nome: Rafael Mesquita de Sousa
Matricula:201602130817
y’’-2y’+y=t^2-1
m^2-2m+1=0
 =1 
yh=c1e^x+c2ce^x
yp=(ax^2+bx+c)-d
y’p=2ax+b
y’’p=2a
2a-2(2ax+b)+(ax^2+bx+c-d)=t^2-1
2a-4ax-2b+ax^2+bx+c-d=t^2-1
(-4a+b)x+92a-2b+c)+ax^2-d=t^2-1
a=1 d=-1
-4a+b=0
2a-2b=0
-4+b=0
2-2b+c=0
B=4
2-2b+c=0
2-2*4+c=0
2-8+c=0
-6+c=
C=6
Yp=(ax^2+bx+c)-d
Yp=1x^2+4x+6-(-1)
Yp=1x^2+4x+7
Yg=c1e^x+c2xe^x+x^2+4x+7
2)y’’-2y’+y=3e^2t
m^2-2m+1=0
 =1 
yh=c1e^x+c2xe^x
yp=ae^2t
y’p=2ae^2t
y’’p=4ae^2t
4ae^2t-2(2ae^2t)+ae^2t=3e^2t
4ae^2t-4ae^2t+ae^2t=3e^2t
A=3
Yg=c1e^x+c2xe^x+3e^2t
3)y’’-2y’+y=4cost
m^2-2m+1=0
 =1 
yh=c1e^x+c2xe^x
yp=accost+bsent
y’p=-asent+bcost
y’’p=-acost-bsent
-acost-bsent-2(-asent+bcost)+(acost+bsent) =4cost
2asent-2bcost=4cost
-bcost=-4cost
A=0 b=-2
Yp=accost+bsent
Yp=0cost-2sent
Yp=-2sent
Yg=c1e^x+c2xe^x-2sent
4)y’’-2y’+y=3e^t
m^2-2m+1=0
 =1 
yh=c1e^x+c2xe^x
yp=at^2e^t
y’p=2ate^t+at^2e^t
y’’p=(2a*e^t+2at*e^t)+(2at*e^t+at^2*e^t)
(2a*e^t+2at*e^t)+(2at*e^t+at^2*e^t)-2(2at*e^t+at^2*e^t)+at^2e^t=3e^t
2ae^t+2ate^t+2ate^t+at^2e^t-4ate^t-2atê^t=3e^t
2ae^t+4ate^t-4ate^t+2at^2e^t-2at^2e^t=3e^t
2ae^t=3e^t
2a=3
A=3/2
Yp=3/2t^2e^t
Yg=c1e^t+c2xe^t+3/2t^2e^t
5)y’’-2y’+y=te^t
m^2-2m+1=0
 =1 
yh=c1e^x+c2xe^x
yp=at^3e^t+bt^2e^t
y’p=3at^2e^t+at^3e^t+2bte^t+bt^2e^t
y’’p=6ate^t+6at^2e^t+at^3e^t+2be^t+4bte^t+bt^2e^t
9ate^t+6at^2e^t+at^3e^t+2be^t+4bte^t+bt^2e^t+6at^2e^t-2at^3e^t-4bte^t-2bt^2e^t+at^3e^t+bt^2e^t=te^t
6ate^t+bt^2e^t=te^t
(6at+2b)e^t=te^t
6ª=1
A=1/6 
2b=0
b=0
yp=1/6t^3e^t
yg=c1e^x+c2xe^t+1/6t^3e^t
6)y’-y=sent+cos2t
m-1=0
m=1
yh=c1e^x+c2xe^x
yp=asent+bcost+csen2t+dcos2t
y’p=accost-bsent+ccos2t-dsen2t
yp= accost-bsent+ccos2t-dsen2t-asent-bcost-csen2t+dcos2t=sent+cos2t
acost+ccos2t-bcost-dcos2t-bsent-dsen2t-asent-csen2t=sent+cost
(a-b)cost+(c-d)cos2t+(-b-a)sent+(-d-csen2t) =sent+cos2t
-a-b=1 *(-1)
A+b=-1
a-b=0 a=b+0
b+b=-1
2b=-1
B=-1/2
a-1/2=-1
a=-1+1/2
a=-1/2
c-d=1
-c-d=0 c=d+0
-d-d=0 
D=0/2
D=0 
c-d=1
c-0=1
c=1
-1/2sent-1/2cost+sen2t
Yg=c1e^t+c2xe^t-1/2sent-1/2cost+sen2t