Question

Show the spherical Bessel functions satisfy the differential equation ${\mathbf{x}}^{\mathbf{2}}{\mathbf{y}}^{\mathbf{\text{'}\text{'}}}\mathbf{+}\mathbf{2}{\mathbf{xy}}^{\mathbf{\text{'}}}\mathbf{+}\left[x^2-n\left(n+1\right)\right]\mathbf{y}\mathbf{=}\mathbf{0}$.

Short answer

It’s proved that spherical Bessel functions satisfy the differential equation

${\mathrm{x}}^2{\mathrm{y}}^{\text{'}\text{'}}+2{\mathrm{xy}}^{\text{'}}+[{\mathrm{x}}^2-\mathrm{n}(\mathrm{n}+1)]\mathrm{y}=0$.

Step-by-step solution

Concept of spherical Bessel's function:

Write the spherical Bessel's function as given below.

${\mathbf{j}}_{\mathbf{n}}\left(x\right)\mathbf{=}\sqrt{\frac{\mathbf{\pi }}{\mathbf{2}\mathbf{x}}{\mathbf{J}}_{\frac{\left(2n+1\right)}{\mathbf{2}}}}\left(x\right)$

Show the spherical Bessel functions satisfy the differential equation  x2y''+2xy'+[x2-n(n+1)]y=0:

Consider the given equation as below.

${\mathrm{x}}^2{\mathrm{y}}^{\text{'}\text{'}}+2{\mathrm{xy}}^{\text{'}}+[{\mathrm{x}}^2-\mathrm{n}(\mathrm{n}+1)]\mathrm{y}=0$

According to the equation,

$y^{\text{'}\text{'}}+\frac{1-2a}{x}{y}^{\text{'}}+\left[{\left(bc{x}^{c-1}\right)}^2+\frac{a^2-p^2{c}^2}{x^2}\right]y=0$ …… (1)

Hence,

$\mathrm{y}={\mathrm{x}}^{\mathrm{a}}{\mathrm{Z}}_{\mathrm{p}}\left({\mathrm{bx}}^{\mathrm{c}}\right)$ …… (2)

Comparing the spherical Bessel's function equation to equation (2) as follows:

$$\begin{gathered} \mathrm{a}=\frac{1}{2} \\ \mathrm{c}=1 \\ \mathrm{p}=\frac{2\mathrm{n}+1}{2} \\ \mathrm{b}=1 \end{gathered}$$

Therefore,

${\mathrm{y}}^{\text{'}\text{'}}+\frac{2}{\mathrm{x}}{\mathrm{y}}^{\text{'}}+\left[1+\frac{{\displaystyle \frac{1}{4}}-{\displaystyle \frac{{\left(2\mathrm{n}+1\right)}^2}{4}}}{{\mathrm{x}}^2}\right]\mathrm{y}=0$

Hence, ${\mathrm{x}}^2{\mathrm{y}}^{\text{'}\text{'}}+2{\mathrm{xy}}^{\text{'}}+2{\mathrm{xy}}^{\text{'}}+\left[{\mathrm{x}}^2-\mathrm{n}\left(\mathrm{n}+1\right)\right]\mathrm{y}=0$.