Expressing Bessel's equation as a matrix system

[Aldiara27]

How do I express Bessel's equation as a matrix system?

This is Bessel's equation:

y'' + (1/t)y' + (1 - (n^2/t^2))y = 0

Thank you for your help in advance!

 

[GeneralSynopsis]

How do I express Bessel's equation as a matrix system?

This is Bessel's equation:

y'' + (1/t)y' + (1 - (n^2/t^2))y = 0

Thank you for your help in advance!

Better late than never? Probably not but here it is anyway:

You set up a state vector for the system:

\(\displaystyle \b{x}=\left[ \begin{array}{c}y\\y'\end{array} \right]\)

Now we write:

\(\displaystyle \b{x}'=f(\b{x})\)

and because this is a linear ODE there exists a matrix \(\displaystyle A\) independent of \(\displaystyle \b{x}\) such that this may be written:

\(\displaystyle \b{x}'=A\b{x}\)

which you should be able to find since:

\(\displaystyle \b{x}'=\left[ \begin{array}{c} y' \\ y'' \end{array} \right]=\left[ \begin{array}{c} \b{x}_2 \\ (n^2/t^2-1)\b{x}_1-\b{x}_2/t \end{array} \right]\)

GS