Chapter 8: Q2P (page 448)

Use L34 and L2 to find the inverse transform of $G (p) H (p)$ whenand $G (p) = 1 / (p + a) and H (p) = 1 / (p + b)$ ; your result should be L7 .

Short Answer

Expert verified

The inverse transforms of is $G (p) H (p) i s \frac{1}{b - a} (e^{- a t} - e^{- b τ}) .$

Step by step solution

Given information.

The given expressions are and $G (p) H (p) i s \frac{1}{(p + a)} a n d H (p) i s \frac{1}{(p + b)}$ .

Inverse transform.

The piecewise-continuous and exponentially-restricted real function f(t) is the inverse Laplace transform of a function F(s), and it has the property:

$L {f} (s) = L {\{f\}} (s) = F (s)$

where L is the Laplace transform.

Find the inverse transform of G(p)H(p). .

Calculate Laplace inverse.

$L_{- 1} (G (p)) H (p) = g * h = \int_{0}^{t} g (t - τ) h (τ) d τ$

Calculate Laplace inverse of $G (p) .$

role="math" localid="1659264162995" $L^{- 1} G (p) = L^{- 1} (\frac{1}{p + a}) g (t) = e^{- a t} g (t - τ) = e^{- a (t - τ)}$

Calculate Laplace inverse of H(p).

$L^{- 1} H (p) = L^{- 1} (\frac{1}{p + b}) h (t) = e^{- b t} h (t - τ) = e^{- b τ}$

Calculate Laplace inverse of G(p)H(p) .

$L^{- 1} [(\frac{1}{p + a}) (\frac{1}{p + b})] = \frac{1}{b - a} (e^{- a t} - e^{- b τ})$

Thus, the Laplace inverse of is $G (p) H (p) i s \frac{1}{b - a} (e^{- a t} - e^{- b τ}) .$ .

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Use L34 and L2 to find the inverse transform of $G (p) H (p)$ whenand $G (p) = 1 / (p + a) and H (p) = 1 / (p + b)$ ; your result should be L7 .

Short Answer

Step by step solution

Given information.

Inverse transform.

Find the inverse transform of G(p)H(p). .

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Use L34 and L2 to find the inverse transform of G(p)H(p)whenand G(p)=1/(p+a)andH(p)=1/(p+b); your result should be L7 .

Short Answer

Step by step solution

Given information.

Inverse transform.

Find the inverse transform of G(p)H(p). .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Modern Physics

Oscillations

Turning Points in Physics

Electricity

Rotational Dynamics

Study anywhere. Anytime. Across all devices.

Use L34 and L2 to find the inverse transform of $G (p) H (p)$ whenand $G (p) = 1 / (p + a) and H (p) = 1 / (p + b)$ ; your result should be L7 .