How do I find the inverse Laplace transform of 4ss2+42?

Name: How do I find the inverse Laplace transform of 4ss2+42?
Uploaded: 2022-02-23T17:22:57+00:00
Description: How do I find the inverse Laplace transform of 4ss2+42?

Hints: Let F(s) be the Laplace transform of f(x), then 1) L{sin⁡(ax)}=aa2+s2 2) L{xf(x)}=ddsF(s)

To find the ILT, you use the definition: f(t)=1i2π∫c−i∞c+i∞dsest4s(s2+4)2 This is an integral in the complex plane; if your are doing these then you are familiar with the technique: define the so-called Bromwich contour and use the residue theorem. That is, f(t)=∑Ress=±2i4sest(s2+4)2 To evaluate the residues at the poles s=±2i, we must evaluate Ress=±2i4sest(s2+4)2=lims⇒±2idds[(s∓2i)24sest(s2+4)2] Doing this is an exercise in taking derivatives and evaluating them at the poles. Note that we are excluding the pieces where the integrand blows up at these poles. f(t)=tsin⁡2t

Hints: Let F(s) be the Laplace transform of f(x), then 1) L{sin⁡(ax)}=aa2+s2 2) L{xf(x)}=ddsF(s)

To find the ILT, you use the definition: f(t)=1i2π∫c−i∞c+i∞dsest4s(s2+4)2 This is an integral in the complex plane; if your are doing these then you are familiar with the technique: define the so-called Bromwich contour and use the residue theorem. That is, f(t)=∑Ress=±2i4sest(s2+4)2 To evaluate the residues at the poles s=±2i, we must evaluate Ress=±2i4sest(s2+4)2=lims⇒±2idds[(s∓2i)24sest(s2+4)2] Doing this is an exercise in taking derivatives and evaluating them at the poles. Note that we are excluding the pieces where the integrand blows up at these poles. f(t)=tsin⁡2t

Key to "How do I find the inverse Laplace transform..."

hvacwk

Answered question

2021-12-08

How do I find the inverse Laplace transform of

{\frac{4 s}{s^{2} + 4}}^{2}

Answer & Explanation

Dabanka4v

Beginner2021-12-09Added 36 answers

Hints: Let F(s) be the Laplace transform of f(x), then
1)

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

L {x f (x)} = \frac{d}{d s} F (s)

Maricela Alarcon

Beginner2021-12-10Added 28 answers

To find the ILT, you use the definition:
$f (t) = \frac{1}{i 2 π} \int_{c - i \infty}^{c + i \infty} d s e^{s t} \frac{4 s}{{(s^{2} + 4)}^{2}}$
This is an integral in the complex plane; if your are doing these then you are familiar with the technique: define the so-called Bromwich contour and use the residue theorem. That is,
$f (t) = \sum R e s_{s = \pm 2 i} \frac{4 s e^{s t}}{{(s^{2} + 4)}^{2}}$
To evaluate the residues at the poles $s = \pm 2 i$ , we must evaluate $R e s_{s = \pm 2 i} \frac{4 s e^{s t}}{{(s^{2} + 4)}^{2}} = lim_{s \Rightarrow \pm 2 i} \frac{d}{d s} [{(s \mp 2 i)}^{2} \frac{4 s e^{s t}}{{(s^{2} + 4)}^{2}}]$
Doing this is an exercise in taking derivatives and evaluating them at the poles. Note that we are excluding the pieces where the integrand blows up at these poles.
$f (t) = t \sin 2 t$

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