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[SOLVED] Homogeneous System

dwsmith

Well-known member
Feb 1, 2012
1,673
$R(r) = A_m\mathcal{J}_m(kr) + B_m\mathcal{Y}_m(kr)$
$$
\begin{pmatrix}
\mathcal{J}_m(ka) & \mathcal{Y}_m(ka)\\
\mathcal{J}_m(kb) & \mathcal{Y}_m(kb)
\end{pmatrix}
\begin{pmatrix}
A_{m}\\
B_{m}
\end{pmatrix} =
\begin{pmatrix}
0\\
0
\end{pmatrix}
$$
In order for our system to have a non-trivial solution, we require that the determinant be 0.
Therefore,
$$
\mathcal{J}_m(ka)\mathcal{Y}_m(kb) - \mathcal{J}_m(kb)\mathcal{Y}_m(ka) = 0
$$
which is our eigenvalue equation.

How do I get to this now:
$$
R(r) = A_m\left[\mathcal{J}_m(kr)-\frac{\mathcal{J}_m(ka)}{\mathcal{Y}_m(ka)} \mathcal{Y}_m(kr)\right]
$$
 
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