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(sorry about the random apostrophe's in some of the writing, i don't want someone to just google and take my working ><. i don't know if the apostrophe thing works though...)

i have two different methods but i don't think either is right, but close. i don't think there is a anything wrong with my calculations, but my problem is more of the form i need to express my answer in.

*i'll try to upload a copy of my handwritten version, to make it easier to read, later tonight*

## Homework Statement

mx" = -kx - cx'

mass = m = 4

Spring constant = k = 1

damp ing constant = c

find the gen'eral solu'tion for the case where the damp'ing cons'tant c = sqrt(7), (under damp ing)

y(t) = Matrix( [y1(t)]

[y2(t)])

in REAL FORM. ( i think the previous line means i have to put it in mtrix form, as two first order e quations)

## Homework Equations

## The Attempt at a Solution

after subbing in the constants into the equation i had:

4x" = -x - sqrt(7)x'

in a previous section i found the matrix to be

A = ( [0 1]

[-1/4 -sqrt(7)/4] )

solving the equation by using the "auxillary equation" method

( 4*lambda^2 + sqrt(7)*lambda + 1 = 0 )

i found lambda (the eigen values) to be:

lambda1 = -sqrt(7)/8 +3i/8

lambda2 = -sqrt(7)/8 -3i/8

here's where my methods split.

->

__METHOD I__

using the underamping equation:

calling little omega "w", and alhpa "a")

lambda1 = -a + wi

lambda2 = -a - wi

(im not sure what happens to the negative when i sub)

A and B are constants

y = A*cos(wt)*e^(-at) + (B*cos(wt)*e^(-at)

y = A*cos((3/8)t)*e^(-(sqrt(7)/8)t) + (B*cos((3/8)t)*e^(-(sqrt(7)/8)t)

this being my gene'ral equation but it needed to be in m'atrix form

i think i need to take that negative i mentioned before into account and use it in my equation so i have the matrix with one positive and one negative

so it would give me:

y = A*cos((3/8)t)*e^(-(sqrt(7)/8)t) + (B*cos((3/8)t)*e^(-(sqrt(7)/8)t)

*method II*

there's too much to write for the second method but:

instead of subbing into the underdamping equation, i sub into the general

y = (C1)*(X1)*e^(lambda1*t) + (C2)*(X2)*e^(lambda2*t)

where matrix X is the eigen vector, i found by using the "A" matrix that i stated at the top of my solution attempt: (A - lambda*I) where "I" is matrix identity for a 2x2 matrix.

***([ ] [ ]) matrix, each set of square brackets represents a new column, each comma separates a row****

X1 = [1], [(sqrt(7)/8) - 3i/8])

X2 = [1], [(sqrt(7)/8) + 3i/8])

lambda1 = -sqrt(7)/8 +3i/8

lambda2 = -sqrt(7)/8 -3i/8 (from previous info given)

i subbed all my new found values into the gener'al solution and got:

y = (C1)*([1], [(sqrt(7)/8) - 3i/8])*e^(((sqrt(7)/8) - 3i/8)*t)

+ (C2)*([1], [(sqrt(7)/8) + 3i/8])*e^((-(sqrt(7)/8) - 3i/8)*t)

with the " e's " and their indexes, i then used the rule:

e^(a+ib) = (e^a)(cos(b) + i sin(b))

and then multiplied them into the matrixes beside them, (as i am following a very similar example) making my equation very very messy and complicated.

after this is where i stopped because i became really confused. the first method is my own i worked out using the help of a textbook. the second is a method of a given example which i think needs to be solved the same to this one.

in the example I am using for method two, they only take the first term of the equation to use the "e" rule on, ... after this part, everything falles apart for me and i can't tell left from right...

*im so sorry there's so much to read, i shortened it as much as i could without making it too confusing (i hope)

any help would be really appreciated.

thankyou for taking the time to read this.