Chapter 12: Q7P (page 593)

(a) Using 15.2 , show that ${鈭�}_{0}^{鈭�} J_{1} (x) dx = 1$ . (b) Use L23of the Laplace Transform Table (Page 469) to show that ${鈭�}_{0}^{鈭�} J_{0} (x) dt = 1$ . (Also see Problem23.29.) $\frac{d}{dx} [x^{- p} J_{p} (x)] = - x^{- p} J_{p + 1} (x)$ .

Short Answer

Expert verified

(a) The required equation is ${鈭�}_{0}^{鈭�} J_{1} (x) dx = 1$ .

(b) Laplace transform equation is ${鈭�}_{0}^{鈭�} J_{0} (t) dt = 1$ .

Step by step solution

Concept of Differential equations and Laplace transform:

Differential equations are equations that connect one or more derivatives of a function. This implies that their answer is a function.

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).

(a) Determine equations and prove:

The given equation is as follow.

$\frac{d}{dx} [{(x)}^{- p} J_{p} (x)] = - {(x)}^{- p} J_{p + 1} (x)$ 鈥�.. (1)

Let, p = 0.

Then,

$\frac{d}{dx} [{(x)}^{o} J_{o} (x)] = - {(x)}^{o} J_{o + 1} (x) \frac{d}{dx} [J_{o} (x)] = - J_{1} (x)$

Integrate from 0 to $鈭�$ as follow.

${鈭�}_{0}^{x} [\frac{d (J_{0} (x))}{dx}] dx = - {鈭�}_{0}^{x} J_{1} (x) dx - {[J_{0} (x)]}^{2} 0 = - {鈭�}_{0}^{x} J_{1} (x) dx {鈭�}_{0}^{x} J_{1} (x) dx = 1$

Therefore,

$(x 鈫� 鈭�, J_{0} (x) 鈫� 0, J_{0} (0) = 1)$

(b) Determining equations with the help of Laplace transform:

The given equation is,

$\frac{d}{dx} [{(x)}^{- p} J_{p} (x)] = - {(x)}^{- p} J_{p + 1} (x)$

Use Laplace transform to calculate the equation as follows:

$L (J_{0} (at)) = \sqrt{p^{2} + a^{2}} = {鈭�}_{0} {(e)}^{- {渭闯}_{0} (at) dt = \sqrt{p^{2} + a^{2}}}$

Therefore,

${鈭�}_{0}^{0} J_{0} (x) dt = 1$

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91影视

(a) Using 15.2 , show that ${鈭�}_{0}^{鈭�} J_{1} (x) dx = 1$ . (b) Use L23of the Laplace Transform Table (Page 469) to show that ${鈭�}_{0}^{鈭�} J_{0} (x) dt = 1$ . (Also see Problem23.29.) $\frac{d}{dx} [x^{- p} J_{p} (x)] = - x^{- p} J_{p + 1} (x)$ .

Short Answer

Step by step solution

Concept of Differential equations and Laplace transform:

(a) Determine equations and prove:

(b) Determining equations with the help of Laplace transform:

One App. One Place for Learning.

Most popular questions from this chapter

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91影视

(a) Using 15.2 , show that 鈭�0鈭�J1(x)dx=1. (b) Use L23of the Laplace Transform Table (Page 469) to show that 鈭�0鈭�J0(x)dt=1. (Also see Problem23.29.) ddx[x-pJpx]=-x-pJp+1(x).

Short Answer

Step by step solution

Concept of Differential equations and Laplace transform:

(a) Determine equations and prove:

(b) Determining equations with the help of Laplace transform:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

Geometrical and Physical Optics

Electricity

Waves Physics

Work Energy and Power

Thermodynamics

Study anywhere. Anytime. Across all devices.

(a) Using 15.2 , show that ${鈭�}_{0}^{鈭�} J_{1} (x) dx = 1$ . (b) Use L23of the Laplace Transform Table (Page 469) to show that ${鈭�}_{0}^{鈭�} J_{0} (x) dt = 1$ . (Also see Problem23.29.) $\frac{d}{dx} [x^{- p} J_{p} (x)] = - x^{- p} J_{p + 1} (x)$ .