A Aldiara27 New member Joined Oct 7, 2008 Messages 4 Oct 28, 2008 #1 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!
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!
G GeneralSynopsis New member Joined Jan 9, 2007 Messages 16 Dec 24, 2008 #2 Aldiara27 said: 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! Click to expand... 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
Aldiara27 said: 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! Click to expand... 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
