UGF
Disciplina: C´
alculo III Profa.: Cl´
audia
QUARTA LISTA DE EXERC´
ICIOS
1) Resolva as equa¸c˜oes:
(a) y00 −8y0+ 7y= 14
(b) y00 +y=x3+ 6
(c) y00 + 2y0+y= 4ex
(d) y00 + 3y0−5y= 0
(e) y00 −7y0+ 12y=−e4x
(f) y00 +y0−6y=xe2x
(g) y00 +y= cos x
(h) y00 −y=x2ex
(i) y00 + 4y=xsen 2x
(j) y00 −2y0+ 10y= sen 3x+ex
(k) y00 −y= 2xsen x
(l) y00 −3y0=x+ cos x
(m) y00 −3y0+ 2y=et+e2t
(n) y00 +y0+ 4y=ex−e−x
(o) y00 −y0−2y= (e2x+e−2x)/2
(p) y00 −4y0+ 4y= 2e2x
(q) 12y00 −5y0−2y= 0
R:y=c1e7x+c2ex+ 2
R:y=c1cos x+c2senx+x3−6x+ 6
R:y=c1e−x+c2xe−x+ex
R:y=c1e(−3+√29)x/2+c2e(−3−√29)x/2
R:y=c1e3x+c2e4x−xe4x
R:y=c1e2x+c2e−3x+xe2xx
10
−
1
25
R:y=c1cos x+c2senx+x
2senx
R:y=c1e−x+c2ex+1
4
−
1
4x+1
6x2xex
R:y=c1cos 2x+c2sen2x+1
16x(sen2x−2xcos 2x)
R:y=ex(c1cos 3x+c2sen3x) + 1
9ex+1
37(sen3x+ 6 cos 3x)
R:y=c1ex+c2e−x−xsenx−cos x
R:y=c1+c2e3x−
1
10 cos x−
3
10senx−
1
6x2−
1
9x
R:y=c1et+c2e2t−tet+te2t
R:y=e−x/2(c1cos √15
2x+c2sen√15
2x) + 1
6ex−
1
4e−x
R:y=c1e2x+c2e−x+1
8e−2x+1
6xe2x
R:y=c1e2x+c2xe2x+x2e2x
R:y=c1e2x/3+c2e−x/4
2) Resolva os problemas de valor inicial:
(a) x00 + 6x0+ 9x=t+ 2
x(0) = 0, x0(0) = 0 R: x(t) = −
4
27
−
5
9te−3t+t
9+4
27
(b) y00 −3y0+ 2y= 0
y(1) = 0, y0(1) = 1 R: y(t) = e2t−2−et−1
(c) x00 + 2x0+ 2x= sen2t+ cos 2t
x(0) = 0, x0(0) = 1 R: x(t) = e−t3
10 cos t+11
10sen t+1
10sen 2t−
3
10 cos 2t
(d) y00 + 16y= 0
y(0) = 2, y0(0) = −2R: y(t) = 2 cos 4t−
sen 4t
2
(e) 2x00 −4x0+ 2x= 0
x(0) = 1, x0(0) = 0 R: x(t) = et−tet
(f) 5y00 +y0=−6t
y(0) = 0, y0(0) = −10 R: y(t) = −200 + 200e−t/5−3t2+ 30t
1
