Find the Laplace transform of f(t)=te−tsin⁡(2t) Then you obtain F(s)=4s+a((s+1)2+4)2 Please type in a = ?

Name: Find the Laplace transform of f(t)=te−tsin⁡(2t) Then you obtain F(s)=4s+a((s+1)2+4)2 Please type in a = ?
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Description: Find the Laplace transform of f(t)=te−tsin⁡(2t) Then you obtain F(s)=4s+a((s+1)2+4)2 Please type in a = ?

Step 1 Given that f(t)=te−tsin⁡(2t) We have to Laplace transform of f(t)=te−tsin⁡(2t) that we have to find L{te−tsin⁡(2t)} Use Laplace transform formula L{tkf(t)}=(−1)kdkdsk(L{f(t)}) For, te−tsin⁡(2t),f(t)=e−tsin⁡(2t) L{te−tsin⁡(2t)}=−ddx(L{e−tsin⁡(2t)})…(1) Now, use formula L{ebtsin⁡(at)}=a(s−b)2+a2 Now, L{e−tsin⁡(2t)}=2(s+1)2+4 Substitute L{e−tsin⁡(2t)}=2(s+1)2+4 in equation (1) L{te−tsin⁡(2t)}=−ddx(2(s+1)2+4) Step 2 Now, L{te−tsin⁡(2t)}=−ddx(2(s+1)2+4) =−(−4(s+1)((s+1)2+4)2) =4s+4((s+1)2+4)2 Therefore, F(s)=4s+4((s+1)2+4)2 Compare it with

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Answered question

2020-11-29

Find the Laplace transform of

f (t) = t e^{- t} \sin (2 t)

Then you obtain

F (s) = \frac{4 s + a}{{({(s + 1)}^{2} + 4)}^{2}}

Please type in a = ?

Answer & Explanation

davonliefI

Skilled2020-11-30Added 79 answers

Step 1
Given that

f (t) = t e^{- t} \sin (2 t)

We have to Laplace transform of

f (t) = t e^{- t} \sin (2 t) that we have to find L {t e^{- t} \sin (2 t)}

Use Laplace transform formula

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

For,

t e^{- t} \sin (2 t), f (t) = e^{- t} \sin (2 t)

L {t e^{- t} \sin (2 t)} = - \frac{d}{d x} (L {e^{- t} \sin (2 t)}) \dots (1)

Now, use formula

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

Now,

L {e^{- t} \sin (2 t)} = \frac{2}{{(s + 1)}^{2} + 4}

Substitute

L {e^{- t} \sin (2 t)} = \frac{2}{{(s + 1)}^{2} + 4}

in equation (1)

L {t e^{- t} \sin (2 t)} = - \frac{d}{d x} (\frac{2}{{(s + 1)}^{2} + 4})

Step 2
Now,

L {t e^{- t} \sin (2 t)} = - \frac{d}{d x} (\frac{2}{{(s + 1)}^{2} + 4})

= - (- \frac{4 (s + 1)}{{({(s + 1)}^{2} + 4)}^{2}})

= \frac{4 s + 4}{{({(s + 1)}^{2} + 4)}^{2}}

Therefore,

F (s) = \frac{4 s + 4}{{({(s + 1)}^{2} + 4)}^{2}}

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