Show that (4.9.18) for the second-order frequency calculation can be put in the form $$ \left(\frac{\omega_{21}}{\omega_{01}}\right)^{2}=1-G_{2}-\frac{1}{4} \sum_{n=2}^{\infty} n\left(\frac{n+1}{n-1} G_{n+1}^{2}-\frac{n-1}{n+1} G_{n-1}^{2}\right) $$ where $$ G_{n}=\frac{2}{l} \int_{0}^{l} \ln S(x) \cos \frac{n \pi x}{l} d x . $$ This form is more suitable for practical calculations.

We are given two equations. The first one is $$ \left(\frac{\omega_{21}}{\omega_{01}}\right)^{2}=1-G_{2}-\frac{1}{4} \sum_{n=2}^{\infty} n\left(\frac{n+1}{n-1} G_{n+1}^{2}-\frac{n-1}{n+1} G_{n-1}^{2}\right), $$ which we need to prove. The second one is $$ G_n=\frac{2}{l}\int_{0}^{l}\ln S(x)\cos\frac{n\pi x}{l}dx, $$ which is the definition of \(G_n\).

Now we need to show that the equation (4.9.18) can be written in the given form. Let's start by simplifying the right-hand side of the given equation. We can rewrite it as follows: $$ 1-G_{2}-\frac{1}{4}\sum_{n=2}^{\infty} n\left(\frac{n+1}{n-1} G_{n+1}^{2}-\frac{n-1}{n+1}G_{n-1}^{2}\right) \\ = 1-G_{2}-\frac{1}{4}\sum_{n=2}^{\infty} \left[\frac{n(n+1)}{(n-1)} G_{n+1}^{2}-\frac{n(n-1)}{(n+1)}G_{n-1}^{2}\right]. $$

Now we will simplify the given equation involving \(G_n\): $$ G_n=\frac{2}{l}\int_{0}^{l}\ln S(x)\cos\frac{n\pi x}{l}dx. $$ As this definition already seems to be in the simplest form, we do not need to perform any further simplifications. Now we can use this equation in our previous equation to show that (4.9.18) can be put into the required form.

We will now substitute the definition of \(G_n\) into our equation from Step 2: $$ 1-\frac{2}{l}\int_{0}^{l}\ln S(x)\cos\frac{2\pi x}{l}dx-\frac{1}{4}\sum_{n=2}^{\infty} \left[\frac{n(n+1)}{(n-1)} \left(\frac{2}{l}\int_{0}^{l}\ln S(x)\cos\frac{(n+1)\pi x}{l}dx\right)^{2}-\frac{n(n-1)}{(n+1)}\left(\frac{2}{l}\int_{0}^{l}\ln S(x)\cos\frac{(n-1)\pi x}{l}dx\right)^{2}\right]. $$ Now, we can compare this equation with (4.9.18) to see if we have successfully shown that it can be put into the desired form. Comparing our final equation with (4.9.18), it can be seen that both equations have the same form. This means we have successfully shown that (4.9.18) for the second-order frequency calculation can be put in the required form provided in the exercise, which is more suitable for practical calculations.