Chapter 7: Q10E (page 404)

Use the convolution theorem to find the inverse Laplace transform of the given function.
$\frac{1}{s^{3} (s^{2} + 1)}$

Short Answer

Expert verified

The inverse Laplace transform for the given function by using the convolution theorem is.

$y (t) = \frac{1}{2} [t^{2} - 2 + 2 \cos t]$

Step by step solution

Step 1: Define convolution theorem

Let $f (t)$ and $g (t)$ be piecewise continuous on $[0, \infty)$ and of exponential order and set $F (s) = L {f} (s) a n d G (s) = L {g} (s)$ , then,

$L {f * g} (s) = F (s) G (s),$
or

$L^{- 1} {F (s) G (s)} (t) = (f * g) (t)$

Find the inverse Laplace transform for the given function

Consider the function,

$\frac{1}{s^{3} (s^{2} + 1)}$

Let, $y (s) = \frac{1}{s^{3} (s^{2} + 1)}$

Take inverse Laplace transform on both sides,

$L^{- 1} [y (s)] = L^{- 1} [\frac{1}{s^{3} (s^{2} + 1)}]$

Use the convolution formula, $L^{- 1} [f (s) \cdot g (s)] = f * g = \int_{0}^{t} f (t - v) g (v) d v$ , where

$f (s) = \frac{1}{s^{2} + 1} a n d f (t) = \sin t$

$g (s) = \frac{1}{s^{3}} a n d g (t) = \frac{1}{2} t^{2}$

Hence, the equation 1 becomes,

$y (t) = \int_{0}^{t} \sin (t - v) \cdot \frac{1}{2} v^{2} d v$

Thus, by using parts rule,

role="math" localid="1664877645923" $y (t) = \frac{1}{2} {[v^{2} \cos (t - v) - \int_{0}^{t} 2 v \cos (t - v) d v]}_{0}^{t} = \frac{1}{2} {[v^{2} \cos (t - v) - 2 [- v \sin (t - v) + \cos (t - v)]]}_{0}^{t} = \frac{1}{2} [t^{2} \cos 0 - 2 [- t \sin 0 + \cos 0 - 0 + 2 [0 + \cos t]]] = \frac{1}{2} [t^{2} - 2 + 2 \cos t]$

Therefore, the inverse Laplace transform for the given function is.

$y (t) = \frac{1}{2} [t^{2} - 2 + 2 \cos t]$

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Use the convolution theorem to find the inverse Laplace transform of the given function.
$\frac{1}{s^{3} (s^{2} + 1)}$

Short Answer

Step by step solution

Step 1: Define convolution theorem

Find the inverse Laplace transform for the given function

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Use the convolution theorem to find the inverse Laplace transform of the given function.1s3(s2+1)

Short Answer

Step by step solution

Step 1: Define convolution theorem

Find the inverse Laplace transform for the given function

One App. One Place for Learning.

Most popular questions from this chapter

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Use the convolution theorem to find the inverse Laplace transform of the given function.
$\frac{1}{s^{3} (s^{2} + 1)}$