exerc_calc2_lista2
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exerc_calc2_lista2


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Exerc´\u131cios de Ca´lculo II- Equac¸o\u2dces Diferenciais Lineares de
Segunda Ordem
1. Descreva o conjunto de soluc¸o\u2dces das seguintes equac¸o\u2dces diferenciais lineares homoge\u2c6neas
(i) y\u2032\u2032 + 4y = 0 (ii) y\u2032\u2032 + 2y\u2032 + y = 0.
(iii) y\u2032\u2032 + y\u2032 \u2212 y = 0. (iv) y\u2032\u2032 \u2212 9y\u2032 + 9y = 0.
(v) y\u2032\u2032 + 2y\u2032 + 4y = 0.
2. Esboce os gra´ficos das func¸o\u2dces satisfazendo
(i)
{
y\u2032\u2032 + 4pi2y = 0,
y(0) = 1, y\u2032(0) = 1. (ii)
{
y\u2032\u2032 + 2y\u2032 + 2y = 0,
y(0) = 0, y\u2032(0) = 1.
(iii)
{
y\u2032\u2032 + 3y\u2032 \u2212 4y = 0,
y(0) = 1, y\u2032(0) = 0. (iv)
{
y\u2032\u2032 \u2212 4y\u2032 + 4y = 0,
y(0) = 0, y\u2032(0) = 1.
3. Ache uma soluc¸a\u2dco particular das seguintes equac¸o\u2dces
(i) y\u2032\u2032 + y\u2032 \u2212 3y = (x+ 3)2. (ii) y\u2032\u2032 \u2212 2y\u2032 + 4y = sen2 x.
(iii) y\u2032\u2032 + y\u2032 \u2212 y = x2ex + cos(3x). (iv) y\u2032\u2032 + 2y\u2032 + y = 2 coshx.
(v) y\u2032\u2032 + y = senx.
4. Resolva os seguintes problemas de valor inicial
(i)
{
y\u2032\u2032 \u2212 4y\u2032 + 13y = (10x\u2212 2)ex,
y(0) = 0, y\u2032(0) = 1.
(ii)
{
y\u2032\u2032 \u2212 2y\u2032 + y = ex cosx,
y(0) = 0, y\u2032(0) = 1.
(iii)
{
y\u2032\u2032 \u2212 3y\u2032 + 3y = 3x2 + 2,
y(0) = 4, y\u2032(0) = 5.
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