Exercícios de calculo 2

Equa¸oes Diferenciais de primeira ordem c˜ Respostas: 1 1. (i) y(x) = (x + 1)4 + C(x + 1)2 , C ∈ IR. 2 (ii) y(x) = e−x (x + C), C ∈ IR. (iii) y(x) = xn (ex + C), C ∈ IR. (iv) y(x) = 1 x + Ce−2e , C ∈ IR. 2 27 −x e3x + x3 − 3x2 + 6x − 6 + e 4 4

(v) s(t) = sen t + C cos t, C ∈ IR. 2. (i) y(x) =

(ii) y(x) = e− sen x (x2 − π 2 ) √ 2 x(x + 1)1/2 2 (iii) y(x) = + x(x + 1)1/2 3 (x − 1)3/2 3 (iv) y(x) = x2 se x ≥ 0 e y(x) = x2 (1 + Ce1/x ) se x < 0, C ∈ IR. 3. (i) y(x) = ln(ex + C), C ∈ IR. (ii) y(x) = (2k − 1) π/2, k ∈ Z e tg y + cos x = C, C ∈ IR. Z (iii) y(x) = ±1 e y(x) = sen (x + C), C ∈ IR (iv) y(x) = 1 e y(x) = (v) cos θ = C, cos φ Cex − 1 , C ∈ IR. Cex + 1

C ∈ IR, C = 0.

(vi) (ln y)2 = ln(x2 ) + C, C ∈ IR. Equa¸˜es diferenciais de segunda ordem co Respostas: 1. (i) y(x) = A cos(2x) + B sen (2x), (ii) y(x) = e−x (A + Bx), A, B ∈ IR. (iii) y(x) = A e (iv) y(x) = A e
( −1− 2
√ 5

A, B ∈ IR.

)x

+B e +B e

( −1+ 2

√ 5

)x

, A, B ∈ IR.

√ ( 9−3 5 )x 2

√ ( 9+3 5 )x 2

, A, B ∈ IR.

1

√ √ (v) y(x) = e−x [A cos( 3x) + B sen ( 3x)], 1 20 107 3 (i) yp (x) = − x2 − x − 3 9 27 (ii) yp (x) = 1 + sen (2x) 8

A, B ∈ IR.

(iii) yp (x) = ex (x2 − 6x + 16) + 1 1 (iv) yp (x) = ex + x2 e−x 4 2 x (v) yp (x) = − cos x 2 4. (i) y(x) = xex

1 [−10 cos(3x) + 3 sen (3x)] 109

(ii) y(x) = ex (1 + x − cos x) √ 3 3x x (iii) y(x) = 2e 2 cos( ) + x2 + 2x + 2 2

2