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.