Chapter 8: Q15P (page 439)

Prove L32 for $n = 1$ . Hint: Differentiate equation (8.1) with respect to $p$ .

Short Answer

Expert verified

L32 for n=1 is $= (- \frac{d}{4 p} [G (p)]}$

Step by step solution

Given information

The given function is $L 32$

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

For $n = 1$

Prove that, $L (t - g (t)) = - \frac{d}{d_{p}} [G (p)]$

Where $G (p) = L (g (t))$

We know that $G (p) = L (g (t))$

$G (p) = \int_{0} g (t) \cdot e^{- p t} d t$

Differentiate with respect to, we get

$\frac{d}{d p} [G (p)] = \frac{d}{d p} [\int_{0}^{\hat{g}} (t) - e^{- p t} d t] = \int_{0}^{ε} g (t) \cdot \frac{\partial}{\partial p} (e^{- m}) d t = \int_{0}^{0} g (t) (- t t^{- p t}) d t = - \int_{0}^{{- p e}^{- p 1 (t g (t) d t}}$

$= - L [tg (t)] = (- \frac{d}{4 p} [G (p)]$

Hence Proved.

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Prove L32 for $n = 1$ . Hint: Differentiate equation (8.1) with respect to $p$ .

Short Answer

Step by step solution

Given information

Definition of Laplace Transformation

Differentiate the given function

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Prove L32 forn=1. Hint: Differentiate equation (8.1) with respect top.

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

Work Energy and Power

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Measurements

Study anywhere. Anytime. Across all devices.

Prove L32 for $n = 1$ . Hint: Differentiate equation (8.1) with respect to $p$ .