Chapter 8: Q26P (page 439)

Use L28 and L4 to find the inverse transform of $p e^{- p t} I (p^{2} + 1)$ .

Short Answer

Expert verified

Answer

The inverse Laplace transforms of is

$L^{- 1} [\frac{p^{- e z}}{p^{2} + 1}] = f (t) = \{\begin{cases} \cos (t - π), \\ 0, \end{cases} \begin{array}{r} t > π \\ t < π \end{array}\}$

Step by step solution

Given information

The given function is $p e^{- p 2} I (p^{2} + 1)$

Definition of Laplace Transformation

A transformation of a function f(x) into the function g(t) that is useful especially in reducing the solution of an ordinary linear differential equation with constant coefficients to the solution of a polynomial equation.

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

Inverse Transformation of the given function

From $(L 28)$

$L [f (t) = \{\begin{cases} g (t - a), \\ 0, \end{cases} \begin{array}{r} t > 0 \\ 0 \end{array}\}] = e^{- p w} G (p)$

From $(L 4)$

$L (\cos f) = \frac{p}{w^{2} + a^{2}}$

From above equation

$L (\cos t) = \frac{p}{p^{2} + 1}$

By use of $(L 28);$

$L [f (t) = \{\begin{array}{l} g (t - a), \\ 0, \end{array} \begin{array}{r} t > a > 0 \\ 0 \end{array}\}] = e^{- p u} G (p)$

And consider the

$L^{- 1} [e^{- p x} G (p)] = f (t) = \{\begin{array}{l} g (t - a), \\ 0, \end{array} \begin{array}{r} t > a > 0 \\ 0 \end{array}\} L^{- 1} [\frac{p x}{p^{2} + 1}] = f (t) = \{\begin{array}{l} g (t - π), \\ 0, \end{array} \begin{array}{r} t > π \\ t < π \end{array}\}$

Thus, the inverse Laplace transforms of $\frac{p \times r}{p^{2} + 1}$ is $L^{- 1} [\frac{p^{- e z}}{p^{2} + 1}] = f (t) = \{\begin{array}{l} \cos (t - π), \\ 0, \end{array} \begin{array}{r} t > π \\ t < π \end{array}\}$ .

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Use L28 and L4 to find the inverse transform of $p e^{- p t} I (p^{2} + 1)$ .

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Inverse Transformation of the given function

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Use L28 and L4 to find the inverse transform ofpe-ptI(p2+1).

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Inverse Transformation of the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physical Quantities And Units

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Study anywhere. Anytime. Across all devices.

Use L28 and L4 to find the inverse transform of $p e^{- p t} I (p^{2} + 1)$ .