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 $π$ .

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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.

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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

Force

Conservation of Energy and Momentum

Medical Physics

Fundamentals of Physics

Physics of Motion

Thermodynamics

Study anywhere. Anytime. Across all devices.

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