Chapter 8: Q25P (page 439)

Use L28 to find the Laplace transform of
$f (t) = {\begin{cases} s i n (t - π / 2), & t > π / 2 . \\ 0, & t < π / 2 . \end{cases}$

Short Answer

Expert verified

Answer

The values of $f (t)$ is $L \{f (t)\} = e^{- \frac{5}{2}} (\frac{1}{p^{2} + 1})$ .

Step by step solution

Given information

The given function is $f (t) = \{\begin{matrix} \sin (t - π / 2), t > π / 2 \\ 0, t < π / 2 \end{matrix}\}$

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)

Differentiate the given function

From $(L 28)$ .

$f (t) = \{\begin{matrix} g (t - a), t > a 0 \\ 0, t < a \end{matrix}\}$

Then,

$L \{f (t)\} = e^{p x} G (p) G (p) = L \{g (t)\}$

Compare the given function with the defined general form ofan obtain.

$a - π / 2 g (t) = \sin t$

L 28)

Result for the given function and substitute the above values.

$L [f (t)] = e^{- p x} G (p) L \{f (t)\} = e^{p w / 2} L (\sin t) L \{f (t)\} = e^{- \frac{π}{2}} (\frac{1}{p^{2} + 1}) L \{\sin a r\} = \frac{a}{p^{2} + α^{2}}$

Thus, the values of $f (f)$ is $L [f (t)] = e^{\frac{m}{2}} (\frac{1}{p^{2} + 1})$ .

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Use L28 to find the Laplace transform of
$f (t) = {\begin{cases} s i n (t - π / 2), & t > π / 2 . \\ 0, & t < π / 2 . \end{cases}$

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Differentiate the given function

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Use L28 to find the Laplace transform off(t)={sint-π/2,t>π/2.0,t<π/2.

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Differentiate the given function

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

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Study anywhere. Anytime. Across all devices.

Use L28 to find the Laplace transform of
$f (t) = {\begin{cases} s i n (t - π / 2), & t > π / 2 . \\ 0, & t < π / 2 . \end{cases}$