"Find the laplace transform of the following: a)t2sin⁡kt..." solved and explained

Name: Find the laplace transform of the following: a)t2sin⁡kt b)tsin⁡kt
Uploaded: 2022-02-23T17:22:57+00:00
Description: Find the laplace transform of the following: a)t2sin⁡kt b)tsin⁡kt

pedzenekO

Answered question

2021-03-05

Find the laplace transform of the following:

a) t^{2} \sin k t

b) t \sin k t

Answer & Explanation

Liyana Mansell

Skilled2021-03-06Added 97 answers

Step 1

Recall the following.

L {t^{k} f (t)} = (- 1)^{k} \frac{d^{k}}{d s^{k}} (L {f (t)})

L {\sin (a t)} = \frac{a}{s^{2} + a^{2}}

Step 2

(a) Obtain the Laplace transform as follows.

(a) f (t) = t^{2} \sin (k t)

L {\sin (k t)} = \frac{k}{s^{2} + k^{2}}

L {t^{2} \sin (k t)} = (- 1)^{2} \frac{d^{2}}{d s^{2}} (L {\sin (k t)})

= \frac{d^{2}}{d s^{2}} (\frac{k}{s^{2} + k^{2}})

= \frac{d}{d s} (\frac{d}{d s} (\frac{k}{s^{2} + k^{2}}))

= \frac{d}{d s} (- \frac{2 k s}{(s^{2} + k^{2})^{2}})

= - 2 k \frac{d}{d s} (\frac{s}{(s^{2} + k^{2})^{2}})

= - 2 k \frac{(s^{2} + k^{2})^{2} \frac{d}{d s} (s) - s \frac{d}{d s} [(s^{2} + k^{2})^{2}]}{((s^{2} + k^{2})^{2})^{2}}

= - 2 k \frac{(s^{2} + k^{2})^{2} - 4 s^{2} [(s^{2} + k^{2})]}{((s^{2} + k^{2})^{2})^{2}}

= - 2 k \frac{(s^{2} + k^{2}) - 4 s^{2}}{(s^{2} + k^{2})^{3}}

= \frac{2 k (3 s^{2} - k^{2})}{(s^{2} + k^{2})^{3}}

Step 3

(b) Obtain the Laplace transform as follows.

(b) f (t) = t \sin (k t)

L \sin (k t) = \frac{k}{s^{2} + k^{2}}

L {t \sin (k t)} = (- 1)^{1} \frac{d^{1}}{d s^{1}} (L {\sin (k t)})

= - \frac{d}{d s} (\frac{k}{s^{2} + k^{2}})

= - k \frac{d}{d s} (\frac{1}{s^{2} + k^{2}})

= - k (\frac{- 1}{(s^{2} + k^{2})^{2}}) \frac{d}{d s} (s^{2} + k^{2})

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