EDOLH de Coeficientes Constantes

Lista 6 - EDOLH de Coeficientes Constantes

1. Encontre a solução geral das sequintes EDOLH’s.

a) y′′ − 3y = 0

b) y′′ − 3y′ = 0

c) y′′′ + 3y′′ = 0

d) y′′′ + 6y′′ + 11y′ + 6y = 0

e) 9y′′′ − 18y′′ − y′ + 2y = 0

f) y(5)

− 3y(4)

− y′′′ + 11y′′ − 12y′ + 4y = 0

g) y′′′ + 3y′ = 0

h) 4y′′ + 8y′ + 5y = 0

2. Resolva as EDOLH’s de ordem superior:

a) y(4)

− 2y′′′ + 2y′ − y = 0

b) y′′′ − y = 0

c) y′′′ + y = 0

3. Encontre a solução geral das sequintes EDOLH’s.

a)  y′′′ + y′ = 0

y(0) = 0; y′(0) = 1; y′′(0) = 2.

b)  y(4) − y = 0

y(0) = 0; y′(0) = 0; y′′(0) = −1; y′′′(0) = 0.

Respostas:

1)

a) y(x) = C1ex√3 + C2e−x√3

b) y(x) = C1 + C2e3x

c) y(x) = C1 + C2x + C3e−3x

d) y(x) = C1e−3x + C2e−2x + C3e−x

e) y(x) = C1e2x + C2ex/3 + C3e−x/3

f) y(x) = C1ex + C2xex + C3x2ex + C4e2x + C5e−2x

g) y(x) = C1 + C2 cos (xp3) + C3 sen (xp3)

h) y(x) = C1e−x cos (x/2) + C2e−x sen (x/2)

em cada item, C1,C2,C3,C4 e C5 2 R.

2)

a) y(x) = Aex + Bxex + Cx2ex + De−x

b) y(x) = Aex + Be−x/2 cos xp3/2 + Ce−x/2 sen xp3/2

c) y(x) = Ae−x + Bex/2 cos xp3/2 + Cex/2 sen xp3/2

em cada item, A,B,C,D 2 R.

3)

a) y(x) = 2 − 2 cos(x) + sen (x)

b) y(x) = −

1

4

ex

−

1

4

e−x +

1

2

cos(x)