Chapter 7: Q6E (page 404)

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

Short Answer

Expert verified

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

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

Step by step solution

Define convolution theorem

Let $f (t)$ and $g (t)$ be piecewise continuous on $[0 \infty)$ and of exponential order $α$ and set,

$F (s) = L {f} (s) a n d G (s) = L {g} (s)$ then,

$L (f * g) (s) = F (s) G (s)$

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

Find the inverse Laplace transform by using the convolution theorem

Consider the given function, $\frac{1}{(s + 1) (s + 2)}$

Let, $F (s) = \frac{1}{(s + 1)}$

$G (s) = \frac{1}{(s + 2)}$

Since, we know

role="math" localid="1665059297605" $f (t) = L^{- 1} \{\frac{1}{s + 1}\} (t) = e^{- t}$

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

Apply inverse Laplace transform and use convolution theorem to obtain,

$L^{- 1} \{\frac{1}{(s + 1) (s + 2)}\} (t) = L^{- 1} {F (s) G (s)} (t) = g (t) * g (t) = e^{- t} e^{- 2 t}$

Use the convolution formula,

$(f * g) (t) = \int_{0}^{t} f (t - v) g (v) d v$

$L^{- 1} \{\frac{1}{(s + 1) (s + 2)}\} (t) = \int_{0}^{t} e^{- (t - v)} e^{- 2 v} d v = e^{- t} \int_{0}^{t} e^{- v} d v = e^{- t} {[- e^{- v}]}_{0}^{t} = e^{- t} (1 - e^{- t})$

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

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

$L^{- 1} \{\frac{1}{(s + 1) (s + 2)}\} (t) = e^{- t} (1 - e^{- t})$ " width="9" height="19" role="math">

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

Short Answer

Step by step solution

Define convolution theorem

Find the inverse Laplace transform by using the convolution theorem

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Use the convolution theorem to find the inverse Laplace transform of the given function.1(s+1)(s+2)

Short Answer

Step by step solution

Define convolution theorem

Find the inverse Laplace transform by using the convolution theorem

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.
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