91Ó°ÊÓ

The recursion relation is given as follows:

$\frac{\mathbf{d}}{\mathbf{dx}}\left(x^{-p}{J}_p\left(x\right)\right)\mathbf{=}\mathbf{-}{\mathbf{x}}^{\mathbf{-}\mathbf{1}}{\mathbf{J}}_{\mathbf{p}\mathbf{+}\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$

As the given equation is,

$Y_{\frac{1}{2}}\left(x\right)=-\sqrt{\frac{2}{\mathrm{Ï€}x}}\cos \left(x\right)$

As you know,

localid="1659330970652" $$\begin{gathered} y_n\left(x\right)=\sqrt{\frac{\mathrm{Ï€}}{2x}}.Y_{\frac{2n+1}{2}\left(x\right)} \\ {y}_n\left(x\right)=-x^n{\left(-\frac{1}{x}\frac{d}{dx}\right)}^n\left(\frac{\cos \left(x\right)}{x}\right)........\left(1\right) \end{gathered}$$

For n = 0,1,2...:

$\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^{-\mathrm{p}}{\mathrm{J}}_{\mathrm{p}}\left(\mathrm{x}\right)\right]=-{\mathrm{x}}^{-\mathrm{p}}{\mathrm{J}}_{\mathrm{p}+1}\left(\mathrm{x}\right)..........\left(2\right)$

From equation (2) obtain:

$$\begin{gathered} \frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{x}\underset{2}{\overset{1}{∑}}{\mathrm{Y}}_{\frac{1}{2}}\left(\mathrm{x}\right)\right]={\mathrm{x}}^{\frac{1}{2}}{\mathrm{Y}}_{\frac{3}{2}}\left(\mathrm{x}\right) \\ \frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^{\frac{3}{2}}{\mathrm{Y}}_{\frac{3}{2}}\left(\mathrm{x}\right)\right]={\mathrm{x}}^{-\frac{3}{2}}{\mathrm{Y}}_{\frac{6}{2}}\left(\mathrm{x}\right) \end{gathered}$$

Therefore,

$$\begin{gathered} {\mathrm{Y}}_{\frac{3}{2}}\left(\mathrm{x}\right)=-{\mathrm{x}}^{\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^{\frac{1}{2}}\left(-\sqrt{\frac{2}{\mathrm{Ï€³\ae }}}\cos \left(\mathrm{x}\right)\right)\right] \\ =-{\mathrm{x}}^{\frac{1}{2}}\frac{\mathrm{d}}{\mathrm{dx}}\left[\frac{1}{\mathrm{x}}\sqrt{\frac{2}{\mathrm{Ï€³\ae }}}\cos \left(\mathrm{x}\right)\right] \\ ={\sqrt{\frac{2}{\mathrm{Ï€³\ae }}}}^{\frac{3}{2}}\left[\sin \left(\mathrm{x}\right)+\mathrm{x}\cos \left(\mathrm{x}\right)\right] \end{gathered}$$

Similarly for ${\mathrm{Y}}_{\frac{5}{2}}\left(\mathrm{x}\right)$:

$$\begin{gathered} {\mathrm{Y}}_{\frac{5}{2}}\left(\mathrm{x}\right)={\mathrm{x}}^{{}_{\frac{3}{2}}}-\frac{\mathrm{d}}{\mathrm{dx}}\left[-\sqrt{\frac{2}{\mathrm{Ï€}}}{\mathrm{x}}^{-\frac{3}{2}}\left(\sin \left(\mathrm{x}\right)+\mathrm{x}\cos \left(\mathrm{x}\right)\right)\right] \\ ={\mathrm{x}}^{{}_{\frac{3}{2}}}-\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^{-3}\sqrt{\frac{2}{\mathrm{Ï€}}}\left(\sin \left(\mathrm{x}\right)+\mathrm{x}\cos \left(\mathrm{x}\right)\right)\right] \\ =-\sqrt{\frac{2}{\mathrm{Ï€}}}{\mathrm{x}}^{-\frac{5}{2}}\left[3\mathrm{x}\sin \left(\mathrm{x}\right)+({\mathrm{x}}^2-3)\cos \left(\mathrm{x}\right)\right] \end{gathered}$$

From equation (1) obtain:

$$\begin{gathered} {\mathrm{Y}}_0\left(\mathrm{x}\right)=\sqrt{\frac{2}{\mathrm{Ï€}}}.{\mathrm{Y}}_{\frac{1}{2}}\left(\mathrm{x}\right) \\ =\sqrt{\frac{2}{\mathrm{Ï€}}}\left(\sqrt{\frac{2}{\mathrm{Ï€³\ae }}}\cos \left(\mathrm{x}\right)\right) \\ =-\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x}} \\ =-{\mathrm{x}}^0{\left(-\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\right)}^0\left(\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x}}\right) \end{gathered}$$

For, n = 0 .

Now,

$$\begin{gathered} {\mathrm{Y}}_{!}\left(\mathrm{x}\right)=\sqrt{\frac{2}{\mathrm{Ï€}}}.{\mathrm{Y}}_{\frac{3}{2}}\left(\mathrm{x}\right) \\ =\sqrt{\frac{2}{2\mathrm{x}}}\left[-\sqrt{\frac{2}{\mathrm{Ï€}}}{\mathrm{x}}^{\frac{3}{2}}\left[\sin \left(\mathrm{x}\right)+\mathrm{x}\cos \left(\mathrm{x}\right)\right]\right] \\ =-{\mathrm{x}}^2\left[\sin \left(\mathrm{x}\right)+\mathrm{x}\cos \left(\mathrm{x}\right)\right] \\ =-{\mathrm{x}}^1{\left(-\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\right)}^1\left(\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x}}\right),\mathrm{for}\mathrm{n}=1 \end{gathered}$$

Similarly for ${\mathrm{y}}_2\left(\mathrm{x}\right)$:

$$\begin{gathered} {\mathrm{Y}}_2\left(\mathrm{x}\right)=\sqrt{\frac{\mathrm{Ï€}}{2\mathrm{x}}}{\mathrm{Y}}_{\frac{5}{2}}\left(\mathrm{x}\right) \\ =\sqrt{\frac{\mathrm{Ï€}}{2\mathrm{x}}}\left[-\sqrt{\frac{2}{\mathrm{Ï€}}}{\mathrm{x}}^{-\frac{5}{2}}\left[3\mathrm{x}\sin \left(\mathrm{x}\right)+({\mathrm{x}}^2-3)\cos \left(\mathrm{x}\right)\right]\right] \\ =-{\mathrm{x}}^{-3}\left[3\mathrm{x}\sin \left(\mathrm{x}\right)+({\mathrm{x}}^2-3)\cos \left(\mathrm{x}\right)\right] \\ ={\mathrm{x}}^2{\left(-\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\right)}^2\left(\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x}}\right),\mathrm{for}\mathrm{n}=2 \end{gathered}$$

With the following formula.

${\left(-\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\right)}^2\left(\frac{\sin \left(\mathrm{x}\right)}{\mathrm{x}}\right)=-\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}{\left(-\frac{1}{\mathrm{x}}\frac{\mathrm{d}}{\mathrm{dx}}\right)}^2\left(\frac{\cos \left(\mathrm{x}\right)}{\mathrm{x}}\right)$

Therefore, you obtain,

$$\begin{gathered} {\mathrm{Y}}_{\frac{3}{2}}\left(\mathrm{x}\right)=-\sqrt{\frac{2}{\mathrm{Ï€}}}{x}^{-\frac{3}{2}}\left[\sin \left(\mathrm{x}\right)+\cos \left(\mathrm{x}\right)\right] \\ {\mathrm{Y}}_{\frac{5}{2}}\left(\mathrm{x}\right)=-\sqrt{\frac{2}{\mathrm{Ï€}}}{x}^{-\frac{5}{2}}\left[3x\sin \left(\mathrm{x}\right)+\left(x^2-3\right)\cos \left(\mathrm{x}\right)\right] \end{gathered}$$