I've got the following question:
The first part of this question I can do. I've deduced that the DE has oscillatory solutions for all $\lambda > 16$; that the Eigenvalues are given by $$ \lambda_{n} = 16 + \frac{n^{2}\pi^{2}}{9} $$ And that the corresponding Eigenfunctions are given by $$ y_{n} = e^{-4x}sin \left( \frac{n\pi x}{3} \right) $$ Further, the equation is expressed in Sturm-Liouville form as $$ \frac{d}{dx} \left[ e^{8x} y' \right] + \lambda e^{8x}y =0 $$
However, I have no real understanding of what the rest of this question is asking - or how to do it. Can anyone please help me with this?
The Sturm-Liouville trick goes like this: start out with the differential equations for eigenfunctions belonging to different eigenvalues $$\frac d{dx}\left[p(x)y_n^{\prime}\right]+\lambda_nq(x)y_n=0$$ $$\frac d{dx}\left[p(x)y_m^{\prime}\right]+\lambda_mq(x)y_m=0$$ Multiply the first equation by $y_m(x)$ and the second by $y_n(x)$ and subtract to get $$\begin{align}&y_m\frac d{dx}\left[p(x)y_n^{\prime}\right]-y_n\frac d{dx}\left[p(x)y_m^{\prime}\right]+(\lambda_n-\lambda_m)q(x)y_my_n\\ &=\frac d{dx}\left[p(x)(y_my_n^{\prime}-y_ny_m^{\prime})\right]+(\lambda_n-\lambda_m)q(x)y_my_n\\ &=0\end{align}$$ Then $$\begin{align}(\lambda_n-\lambda_m)\int_a^bq(x)y_m(x)y_n(x)dx&=-\int_a^b\frac d{dx}\left[p(x)(y_m(x)y_n^{\prime}(x)-y_n(x)y_m^{\prime}(x))\right]\\ &=-\left[p(x)(y_m(x)y_n^{\prime}(x)-y_n(x)y_m^{\prime}(x))\right]_a^b\\&=0\end{align}$$ given suitable boundary conditions. Here $y(0)=y(3)=0$, so you know that $$\left(\frac{n^2\pi^2}9-\frac{m^2\pi^2}9\right)\int_0^3e^{8x}y_n(x)y_m(x)dx=0$$ So then $$\int_0^3e^{8x}y_n(x)y_m(x)dx=0$$ if $n\ne m$. Now explicitly if $n\ne m$, $$\begin{align}\int_0^3e^{8x}y_n(x)y_m(x)dx&=\int_0^3e^{8x}e^{-4x}\sin\left(\frac{n\pi x}3\right)e^{-4x}\sin\left(\frac{m\pi x}3\right)dx\\ &=\frac12\int_0^3\left[\cos\left(\frac{(n-m)\pi x}3\right)-\cos\left(\frac{(n+m)\pi x}3\right)\right]dx\\ &=\frac12\left[\frac3{(n-m)\pi}\sin\left(\frac{(n-m)\pi x}3\right)-\frac3{(n+m)\pi}\sin\left(\frac{(n+m)\pi x}3\right)\right]_0^3\\ &=0\end{align}$$