T4 - Applied Mathematical Analysis MATH2100 PDF

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Applied Mathematical Analysis MATH2100...

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MATH2100 Problem Sheet 4

Question 1 Find the inverse Laplace transform of the following functions: a)

c)

25 (s − 1)2 (s2 + 4)

b)

s + 2s + 1

d)

s+2 (s − 1)(s − 2)s

f)

2s2

e) g)

s(s2 s2

1 + 4)2

s+1 − 4s + 5

6s2 + 9s + 1 (s + 1)2 (s − 1)

s (s2 − 4s + 20)(s − 1)

Question 2 Solve the following initial value problems using Laplace transforms: a) y˙ − 3y = −2e3t − 2et + 4e−t , y(0) = 0 b) y˙ − 3y = t(t + 2)e3t ,

y(0) = 2

c) y¨ + 2y = 0,

y0 = c1 , y(0) ˙ = c2

d) y¨ − 6y˙ + 9y = te3t ,

y(0) = 1, y(0) ˙ =0 .

Question 3 Use Laplace transforms to solve the following initial value problems: a) y¨(t) + 2y˙ (t) − 15y(t) = −60e−2t , b) c)



y˙ 1 (t) y˙ 2 (t)



=



y(0) = 2, y(0) ˙ = 26

−y1 + 2y2 + 2e−3t cos t − 72 e−3t sin t −2y1 − y2

y¨1 (t) + 2y2 (t) = et , y˙ 2 (t) = y1 (t) + y2 (t)



, y(0) =



0 0



y1 (0) = y˙ 1 (0) = y2 (0) = 0 .

Question 4 The inverse Laplace transform of repeated irreducible quadratic factors can be found using the result L(t f (t)) = −

dF (s) . ds

Use this result to find the inverse Laplace transform of the following: a)

(s2

2s + 4)2

b)

1

(s2

1 . + 4)2 s

Question 5 Use the convolution theorem to find the inverse Laplace transform of the following functions: a) H(s) =

1 s2 (s + 2)

b) H(s) =

e−3s . (s + 1)(s + 2)

Question 6 Use the second shifting theorem to find the Laplace transform of the following functions: a) f (t) =



4 − t2 , t < 2 0 , t≥2

b) f (t) =

2



t2 , t < 4 . t , t≥4...