How to solve this system of three coupled differential equations

Question

I need to integrate this system of partial derivatives:

\begin{align} x' - y' + z = 0 \\ y' - z' + x = 0 \\ z' - x' + y = 0 \end{align}

Answer 1

Guess the form: $x(t) = a_x \cos(b_x t) + c_x \sin (d_x t)$ and likewise for $y(t)$ and $z(t)$. Perform your differentiation and equate coefficients to find:

$$x(t) = \sqrt{3} \left(c_2-2 c_1\right) \sin \left(\frac{t}{\sqrt{3}}\right)+3 c_2 \cos \left(\frac{t}{\sqrt{3}}\right)$$

$$y(t) = \sqrt{3} \left(c_1-2 c_2\right) \sin \left(\frac{t}{\sqrt{3}}\right)-3 c_1 \cos \left(\frac{t}{\sqrt{3}}\right)$$

$$z(t) = \sqrt{3} \left(c_1+c_2\right) \sin \left(\frac{t}{\sqrt{3}}\right)+3 \left(c_1-c_2\right) \cos \left(\frac{t}{\sqrt{3}}\right)$$