Chapter 12: Q5P (page 592)

To sketch the graph of $\sqrt{x} J_{\frac{1}{2}} (x)$ for x from 0 to $π$ .

Short Answer

Expert verified

The graph of the function $\sqrt{x} J_{\frac{1}{2}} (x)$ is given below:

Step by step solution

Step 1: Concept of Bessel’s Equation:

The solution of Bessel's equation is, $x^{2} y " + xy' + (x^{2} - n^{2}) y = 0$ .

$J_{n} (x) = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(k + 1) ⌜ (n + k + 1)} {(\frac{x}{2})}^{2 k + n}$

Calculation of the function xJ12(x) for x :

Given function $J_{p} (x)$ for $p = \frac{1}{2}$ and x from 0 to $π$ .

role="math" localid="1659269941142" $J_{p} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(n + 1) ⌜ (n + 1 + p)} {(\frac{x}{2})}^{2 n + p}$

For, $p = \frac{1}{2}$ in above function.

$J_{p} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(n + 1) ⌜ (n + 1 + p)} {(\frac{x}{2})}^{2 n + p} \sqrt{x} J_{\frac{1}{2}} (x) = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(n!) (n + \frac{1}{2})!} {(\frac{x}{2})}^{2 n + \frac{1}{2} + \frac{1}{2}} (∵ ⌜ (n + 1) = n!) \sqrt{x} J_{\frac{1}{2}} (x) = \frac{1}{(\frac{1}{2})!} (\frac{x}{2}) - \frac{1}{(\frac{3}{2})!} {(\frac{x}{2})}^{3} + (\frac{1}{2!}) \frac{1}{(\frac{5}{2})!} {(\frac{x}{2})}^{5} - (\frac{1}{3!}) \frac{1}{(\frac{7}{2})!} {(\frac{x}{2})}^{7} + (\frac{1}{4!}) \frac{1}{(\frac{9}{2})!} (\frac{x}{2}) = {(\frac{1}{5!})}^{2} - \frac{1}{(\frac{11}{2})!} {(\frac{x}{2})}^{11} + (\frac{1}{6!}) - \frac{1}{(\frac{13}{2})!} {(\frac{x}{2})}^{13} - (\frac{1}{7!}) - \frac{1}{(\frac{15}{2})!} {(\frac{x}{2})}^{15} + . . . . . .$

Draw the graph of the function xJ12(x) :

The graph of the function $\sqrt{x} J_{\frac{1}{2}} (x)$ is given below:

From the above graph, we can observe that there are three zeros of $\sqrt{x} J_{\frac{1}{2}} (x)$ from $x = 0$ to $π$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

To sketch the graph of $\sqrt{x} J_{\frac{1}{2}} (x)$ for x from 0 to $π$ .

Short Answer

Step by step solution

Step 1: Concept of Bessel’s Equation:

Calculation of the function xJ12(x) for x :

Draw the graph of the function xJ12(x) :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

Electrostatics

Fields in Physics

Kinematics Physics

Circular Motion and Gravitation

Turning Points in Physics

Study anywhere. Anytime. Across all devices.

To sketch the graph of xJ12(x)for x from 0 to π.

Short Answer

Step by step solution

Step 1: Concept of Bessel’s Equation:

Calculation of the function xJ12(x) for x :

Draw the graph of the function xJ12(x) :

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

Electrostatics

Fields in Physics

Kinematics Physics

Circular Motion and Gravitation

Turning Points in Physics

Study anywhere. Anytime. Across all devices.

To sketch the graph of $\sqrt{x} J_{\frac{1}{2}} (x)$ for x from 0 to $π$ .