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## Homework Statement

## Homework Equations

Laplace Transforms

## The Attempt at a Solution

Using basic physics knowledge I got

m

_{1}a

_{1}=-k

_{1}x

_{1}+k

_{2}(x

_{2}-x

_{1})

and

m

_{2}a

_{2}=-k

_{3}x

_{2}-k

_{2}(x

_{2}-x

_{1})

Sub in values and use laplace transforms and rearrange partial fraction and I found that

By doing this I am assuming that the x

^{II}and x

^{III}will equal 0 when t=0 because this is not stated at all in the question. do you believe this is correct?

x

_{1}=-(1/3)cos2t-(2/3)sin2t+(4/3)cost+(8/3)sint

x

_{2}=-(2/3)cos2t+(1/3)sin2t+(8/3)cost-(4/3)sint

Finding the initial displacements by subbing in t=0 for both

_{x_1}and

_{x_2}comes out with what is written in the question 1 and 2, respectively.

However, when I try to sub t=0 into the differentials of the 2 above equations. I believe I should receive the initial velocities stated in the question. however I do not receive these results.

I receive 4/3 for x

_{1}and -2/3 for x

_{2}

the difference between these values and the actual values appears to differ once I differentiate the sin2t and it is multiplied by 2

Does anyone know if I should receive the values listed in the question using this methods and have just made a calculation error in my working earlier on, or should I have done something differently.

Also, the next part of the questions asks to use a matrix and eigenvalues/eigenvectors to solve it. any pointers to help me get started

Thanks very much