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This question already has an answer here:

Consider the coupled spring-mass system with a frictionless table, two masses $m_1$ and $m_2$, and three springs with spring constants $k_1, k_2$, and $k_3$ respectively. The equation of motion for the system are given by:

$y_1''=-\frac{(k_1+k_2)}{m_1}y_1+\frac{k_2}{m_1}y_2$

$y_2''=\frac{k_2}{m_2}y_1-\frac{(k_1+k_2)}{m_2}y_2$

Assume that the masses are $m_1 = 2$, $m_2 = 9/4$, and the spring constants are $k_1=1,k_2=3,k_3=15/4$.

a)use 4x4 system of first order equations to model this system of two second order equations. (hint: $x_1=y_1,x_2=y_2,x_3=y_1',x_4=y_2'$)

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marked as duplicate by LutzL differential-equations Dec 5 '18 at 12:35

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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Introducing a whole new set of $x_i$ variables is somewhat confusing. Simpler to just introduce two new variables $y_3$ and $y_4$ where $y_3=y_1'$ and $y_4=y_2'$. So you now have:

$y_1' = y_3$

$y_2' = y_4$

$y_3' = y_1'' = \dots$

$y_4'=y_2''= \dots$

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