Finding equations of motion of multi degree freedom system

[Setareh7796]

Homework Statement

I am not sure where I am going wrong because I am not getting the right answer.

Relevant Equations

mz̈ + kz = 0

 

[jambaugh]

$$\frac{2k^2}{2k+k} = \frac{2}{3}k$$
in your first line of your second matrix:
$$3k+ \frac{2}{3}k = \frac{9}{3}k + \frac{2}{3}k = \frac{11}{3}k$$
so you're good there once you do the arithmetic.
second line... hmmm.
$$-\frac{2k^2}{2k+k}x_1 +\left(\frac{2k^2}{2k+k} + \frac{4k^2}{4k+k} \right)x_2$$
$$\frac{2}{3}k + \frac{4}{5}k = \frac{10+12}{15}k = \frac{22}{15}k$$
I see no problems except that you're not combining like terms but instead writing those terms as separate matrix entries.