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Exercise 6

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Exercise 6

1. In each of the following functions, express the function in terms of the Heaviside function and find the Laplace transform of the function.

(a) f(t) =

(1 06t <7 cost t >7 (b) f(t) =

(t 06t <3

1−3t t >3

(c) f(t) =





−4 06t <1 0 1< t <3 e−t t >3

2. Find the inverse Laplace transform for each of the following functions.

(a) 1

s2+ 4s+ 12.

(b) 1

s2−4s+ 5 (c) e−5s

s3 (d) e−2s

s2+ 9 (e) s−4

s2−8s+ 10

(f) 1

s(s2+ 16)e−21s 3. Solve the following IVPs.

(a) y00+ 4y =f(t) =

(0 06t <4

3 t >4 , y(0) = 1, y0(0) = 0.

(b) y00+ 5y0+ 6y=f(t) =

(−2 06t <3

0 t >3 , y(0) = 0, y0(0) = 0 (c) y00−4y0+ 4y =f(t) =

(t 06t <3

t+ 2 t >3 , y(0) =−2, y0(0) = 1 (d) y00+ 4y0+ 5y= deg(t−1), y(0) = 0, y0(0) = 3

(e) y00+ 3y0+ 2y= 10(sint+δ(t−1)), y(0) = 1,y0(0) =−1

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