Free oscillations and natural frequencies question

Question

I'm given a mass-and-spring system with a couple second-order differential equations describing the behavior of the system. Those are:

$$x^{\prime \prime} +50x - \frac{25}{2}y = 0$$ and $$y^{\prime \prime} + 50y -50x = 0$$

We set up an operational determinant and calculate the general solutions as follows, as well as make sure we have four independent arbitrary constants

$$x(t) = a_1 \cos (5t) + a_2 \sin (5t) + a_3 \cos (5t \sqrt{3}) + a_4 \sin (5t \sqrt {3})$$ $$y(t) = 2a_1 \cos (5t) + 2a_2 \sin (5t) - 2a_3 \cos (5t \sqrt{3}) - 2a_4 \sin (5t \sqrt {3})$$

Using a bit of trigonometry, we can rewrite the equations a bit in terms of two particular solutions $(x_1, y_1)$ and $(x_2, y_2)$ of the system for

$$x_1 (t) = a_1 \cos (5t) + a_2 \sin (5t) = A \cos(5t - \alpha)$$ $$y_1 (t) = 2a_1 \cos (5t) + 2a_2 \sin (5t) = 2A \cos(5t - \alpha)$$ and $$x_2 (t) = a_3 \cos (5t \sqrt{3}) + a_4 \sin (5t \sqrt {3}) = B \cos(5t\sqrt{3} - \beta)$$ $$y_2 (t) = -2a_3 \cos (5t \sqrt{3}) - 2a_4 \sin (5t \sqrt {3}) = -2B \cos(5t\sqrt{3} - \beta)$$

Here are my questions (I need a really intuitive answer because I don't really understand this):

#1. How do we know that those two solutions, $(x_1, y_1)$ and $(x_2, y_2)$, are actually solutions? More specifically, what is the significance about these solutions? Obviously we can reach the solutions when we set $a_3$ and $a_4$ to zero and then do the same with $a_1$ and $a_2$... but what's the intuition behind this?

**#2. The book describes these two solutions as natural modes of oscillation that exhibit two natural frequencies. What are natural modes of oscillation and how do I find the natural frequencies? Could someone give me a physical description of this answer? **

Thank you in advance.

Answer 1

1a) You can check that $(x_1,y_1)$ and $(x_2,y_2)$ are solutions by substituting them in the system of ODEs, as they should solve the system, i.e. they should obey the equations.

1b) and 2. Your solutions are linear combinations of the functions $\cos(5t)$, $\sin(5t)$, $\cos(5\sqrt{3}\,t)$ and $\sin(5\sqrt{3}\,t)$. Note that these functions are of the form $\cos/\sin(\alpha\,t)$, with $\alpha = 5$ or $\alpha = 5 \sqrt{3}$. The function $\cos(\alpha t)$ describes an oscillation with period $\frac{2\pi}{\alpha}$, i.e. with frequency $\frac{\alpha}{2\pi}$, see here. The same holds for $\sin(\alpha t)$. Therefore, you see that the general solution to the system of ODEs combines oscillations with two separate frequencies, $\frac{5}{2\pi}$ and $\frac{5\sqrt{3}}{2\pi}$. By choosing either $a_{1,2}=0$ or $a_{3,4}=0$, you can find two solutions, with $(x_1,y_1)$ only exhibiting oscillations with frequency $\frac{5}{2\pi}$, and $(x_2,y_2)$ only exhibiting oscillations with frequency $\frac{5\sqrt{3}}{2\pi}$.

Now, you can see that the general solution $(x,y)$ is a superposition (i.e. a linear combination) of these two solutions $(x_1,y_1)$ and $(x_2,y_2)$.