Chapter 12: Q9P (page 593)

Use L23of the Laplace Transform Table (page 469 ) to evaluate $\int_{0}^{\infty} {tJ}_{0} (2 t) e \hat{} - tdt$ .

Short Answer

Expert verified

The equation by use of Laplace theorem is $\int_{0}^{\infty} [({tJ}_{0}) (2 t)] e^{- 1} dt = \sqrt{5}$ .

Step by step solution

Concept of Laplace Transform

The Laplace transform is an integral transform in mathematics that turns a function of a real variable (typically time) to a function of a complex variable. It is named after its inventor Pierre-Simon Laplace. (Complex periodicity)

Determine equations with proof:

The given equation is as follow.

$\int_{0}^{\infty} [{tJ}_{0} (2 t)] e^{- 1} dt$ ….. (1)

Let $g (t) = ({tJ}_{0}) - (2 t)$

Therefore,

$\int_{0}^{\infty} [{tJ}_{0} (2 t)] e^{- 1} dt = L (g (t))$

Thus,

$L (g (t)) = L [({tJ}_{0}) (2 t)] = - \frac{d}{dp} L [\frac{1}{\sqrt{P^{2} + 4}}] = p \sqrt{P^{2} + 4}$

At p = 1

$L (g (t)) = 1 \times \sqrt{{(1)}^{2} + 4} = \sqrt{5}$

Hence, the solution is role="math" localid="1659263208798" $\int_{0}^{\infty} [({tJ}_{0}) (2 t)] e^{- 1} dt = \sqrt{5}$ .

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Use L23of the Laplace Transform Table (page 469 ) to evaluate $\int_{0}^{\infty} {tJ}_{0} (2 t) e \hat{} - tdt$ .

Short Answer

Step by step solution

Concept of Laplace Transform

Determine equations with proof:

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Use L23of the Laplace Transform Table (page 469 ) to evaluate ∫0∞tJ0(2t)e^-tdt.

Short Answer

Step by step solution

Concept of Laplace Transform

Determine equations with proof:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

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Study anywhere. Anytime. Across all devices.

Use L23of the Laplace Transform Table (page 469 ) to evaluate $\int_{0}^{\infty} {tJ}_{0} (2 t) e \hat{} - tdt$ .