Question

To sketch the graph of $\sqrt{\mathbf{x}}{\mathbf{J}}_{\frac{\mathbf{1}}{\mathbf{2}}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$for x from 0 to $\mathbf{\pi }$.

Short answer

The graph of the function $\sqrt{\mathrm{x}}{\mathrm{J}}_{\frac{1}{2}}\left(\mathrm{x}\right)$is given below:

Step-by-step solution

 Step 1: Concept of Bessel’s Equation:

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

${\mathbf{J}}_{\mathbf{n}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{\sum }_{\mathbf{k}\mathbf{=}\mathbf{0}}^{\mathbf{\infty }}\frac{{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{k}}}{\mathbf{(}\mathbf{k}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{⌜}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{k}\mathbf{+}\mathbf{1}\mathbf{)}}{\mathbf{\left(}\frac{\mathbf{x}}{\mathbf{2}}\mathbf{\right)}}^{\mathbf{2}\mathbf{k}\mathbf{+}\mathbf{n}}$

Calculation of the function xJ12(x) for x :

Given function ${\mathrm{J}}_{\mathrm{p}}\left(\mathrm{x}\right)$for $\mathrm{p}=\frac{1}{2}$and x from 0 to $\mathrm{\pi }$.

role="math" localid="1659269941142" ${\mathrm{J}}_{\mathrm{p}}\left(\mathrm{x}\right)=\sum _{\mathrm{n}=0}^{\infty }\frac{{(-1)}^{\mathrm{n}}}{(\mathrm{n}+1)⌜(\mathrm{n}+1+\mathrm{p})}{\left(\frac{\mathrm{x}}{2}\right)}^{2\mathrm{n}+\mathrm{p}}$

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

$$\begin{gathered} {\mathrm{J}}_{\mathrm{p}}\left(\mathrm{x}\right)=\sum _{\mathrm{n}=0}^{\infty }\frac{{(-1)}^{\mathrm{n}}}{(\mathrm{n}+1)⌜(\mathrm{n}+1+\mathrm{p})}{\left(\frac{\mathrm{x}}{2}\right)}^{2\mathrm{n}+\mathrm{p}} \\ \sqrt{\mathrm{x}}{\mathrm{J}}_{\frac{1}{2}}\left(\mathrm{x}\right)=\sum _{\mathrm{n}=0}^{\infty }\frac{{\left(-1\right)}^{\mathrm{n}}}{(\mathrm{n}!)\left(\mathrm{n}+{\displaystyle \frac{1}{2}}\right)!}{\left(\frac{\mathrm{x}}{2}\right)}^{2\mathrm{n}+\frac{1}{2}+\frac{1}{2}}\left(\because ⌜(\mathrm{n}+1)=\mathrm{n}!\right) \\ \sqrt{\mathrm{x}}{\mathrm{J}}_{\frac{1}{2}}\left(\mathrm{x}\right)=\frac{1}{\left({\displaystyle \frac{1}{2}}\right)!}\left(\frac{\mathrm{x}}{2}\right)-\frac{1}{\left({\displaystyle \frac{3}{2}}\right)!}{\left(\frac{\mathrm{x}}{2}\right)}^3+\left(\frac{1}{2!}\right)\frac{1}{\left({\displaystyle \frac{5}{2}}\right)!}{\left(\frac{\mathrm{x}}{2}\right)}^5-\left(\frac{1}{3!}\right)\frac{1}{\left({\displaystyle \frac{7}{2}}\right)!}{\left(\frac{\mathrm{x}}{2}\right)}^7+\left(\frac{1}{4!}\right)\frac{1}{\left({\displaystyle \frac{9}{2}}\right)!}\left(\frac{\mathrm{x}}{2}\right) \\ ={\left(\frac{1}{5!}\right)}^2-\frac{1}{\left({\displaystyle \frac{11}{2}}\right)!}{\left(\frac{\mathrm{x}}{2}\right)}^{11}+\left(\frac{1}{6!}\right)-\frac{1}{\left({\displaystyle \frac{13}{2}}\right)!}{\left(\frac{\mathrm{x}}{2}\right)}^{13}-\left(\frac{1}{7!}\right)-\frac{1}{\left({\displaystyle \frac{15}{2}}\right)!}{\left(\frac{\mathrm{x}}{2}\right)}^{15}+...... \end{gathered}$$

Draw the graph of the function xJ12(x) :

The graph of the function $\sqrt{\mathrm{x}}{\mathrm{J}}_{\frac{1}{2}}\left(\mathrm{x}\right)$ is given below:

From the above graph, we can observe that there are three zeros of $\sqrt{\mathrm{x}}{\mathrm{J}}_{\frac{1}{2}}\left(\mathrm{x}\right)$from$x=0$to $\pi$.