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$$

X^{(4)}_n - \alpha^4X_n = 0\qquad\qquad X^{(4)}_m - \alpha^4X_m = 0

$$

\begin{alignat*}{7}

X(0) & = & 0 &\qquad\qquad & X'(0) & = & 0\\

X(L) & = & 0 &\qquad\qquad & X''(L) & = & 0

\end{alignat*}

Using a Sturm-Liouville methodology:

Show that the these two equations can be combined in the following manner

$$

X^{(4)}_nX_m - X^{(4)}_mX_n + (\alpha^4_n - \alpha_m^4)X_nX_m = 0

$$

I am not sure how to start this.

X^{(4)}_n - \alpha^4X_n = 0\qquad\qquad X^{(4)}_m - \alpha^4X_m = 0

$$

\begin{alignat*}{7}

X(0) & = & 0 &\qquad\qquad & X'(0) & = & 0\\

X(L) & = & 0 &\qquad\qquad & X''(L) & = & 0

\end{alignat*}

Using a Sturm-Liouville methodology:

Show that the these two equations can be combined in the following manner

$$

X^{(4)}_nX_m - X^{(4)}_mX_n + (\alpha^4_n - \alpha_m^4)X_nX_m = 0

$$

I am not sure how to start this.

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