Laplace Transforms

  • Thread starter jake96
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  • #26
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No. There is only one answer. It's a linear problem.

Chet
so the answers we have been talking about should be the correct ones (laplace then algebra) cos the first time I used algebra before laplace and that resulted in x''(0) and x'''(0) values which are not stated
 
  • #27
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Yes. I made an algebra error in the last step. When I corrected it, I got your result. I think the mistake that Jake made was inverting ##-\frac{1}{s^2+4}##. The inverse of this is ##-\frac{1}{2}\sin 2t##, not -sin 2t.

Chet
I see my error now, I had 1 over instead of 2 over which will cause the factor to be 1/2.

thanks very much
 
  • #28
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Would you have any pointers to help me get started on doing this problem with a matrix using eigenvalues and eigenvectors

I can't seem to see where the matrix will come in for this question. I'm confident I can solve the eigenvalues and vectors once I've gotten out the matrix.

Thanks again
 
  • #29
vela
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so the answers we have been talking about should be the correct ones (laplace then algebra) cos the first time I used algebra before laplace and that resulted in x''(0) and x'''(0) values which are not stated
Now that you know the correct answers, you can actually calculate those derivatives at t=0, and you'll see that you can't justify setting them to 0 when solving the fourth-order DE using Laplace.
 
  • #30
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OK Jake,

Suppose you had a set of 4 coupled first order linear homogeneous ODEs in 4 unknowns, y1,...,y4. Would you know how to obtain the complementary solution to that set of equations using eigenvalues and eigenvectors?

Chet
 
  • #31
Ray Vickson
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Homework Statement


tFewRWs.png


Homework Equations


Laplace Transforms

The Attempt at a Solution



Using basic physics knowledge I got
m1a1=-k1x1+k2(x2-x1)
and
m2a2=-k3x2-k2(x2-x1)

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

By doing this I am assuming that the xII and xIII will equal 0 when t=0 because this is not stated at all in the question. do you believe this is correct?


x1=-(1/3)cos2t-(2/3)sin2t+(4/3)cost+(8/3)sint
x2=-(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 x1 and -2/3 for x2
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
I'll just give you a hint for the next part. Let ##y_1 = x_1##, ##y_2 = dx_1/dt## (velocity of 1), ##y_3 =x_2## and ##y_4 = dx_2/dt ## (velocity of 2). Your DE system can be written as
[tex] \begin{array}{rcl}dy_1 /dt &=& y_2 \\
dy_2/dt &=& -3 y_1 + 2 y_3 \\
dy_3/dt &=& y_4\\
dy_4/dt &=& y_1 - 2 y_3\end{array}
[/tex]
This can be written as
[tex] \frac{d}{dt} \pmatrix{y_1\\y_2\\y_3\\y_4}=
\pmatrix{0&1&0&0\\-3&0&2&0\\0&0&0&1\\1 &0 -2&0} \pmatrix{y_1\\y_2\\y_3\\y_4} [/tex]
This is of the form ##dY/dt = A Y##, where ##Y## is the column vector of ##y_i##s and ##A## is the ##4 \times 4## matrix above. The solution is of the form
[tex] Y(t) = \exp(A t) Y(0) [/tex]
You can compute the matrix exponential from the eigenvalues and eigenvectors (at least, if none of the eigenvalues are repeated). Google 'matrix exponential'.
 
  • #32
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I'll just give you a hint for the next part. Let ##y_1 = x_1##, ##y_2 = dx_1/dt## (velocity of 1), ##y_3 =x_2## and ##y_4 = dx_2/dt ## (velocity of 2). Your DE system can be written as
[tex] \begin{array}{rcl}dy_1 /dt &=& y_2 \\
dy_2/dt &=& -3 y_1 + 2 y_3 \\
dy_3/dt &=& y_4\\
dy_4/dt &=& y_1 - 2 y_3\end{array}
[/tex]
This can be written as
[tex] \frac{d}{dt} \pmatrix{y_1\\y_2\\y_3\\y_4}=
\pmatrix{0&1&0&0\\-3&0&2&0\\0&0&0&1\\1 &0 -2&0} \pmatrix{y_1\\y_2\\y_3\\y_4} [/tex]
This is of the form ##dY/dt = A Y##, where ##Y## is the column vector of ##y_i##s and ##A## is the ##4 \times 4## matrix above. The solution is of the form
[tex] Y(t) = \exp(A t) Y(0) [/tex]
You can compute the matrix exponential from the eigenvalues and eigenvectors (at least, if none of the eigenvalues are repeated). Google 'matrix exponential'.
thanks
 
  • #33
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I'll just give you a hint for the next part. Let ##y_1 = x_1##, ##y_2 = dx_1/dt## (velocity of 1), ##y_3 =x_2## and ##y_4 = dx_2/dt ## (velocity of 2). Your DE system can be written as
[tex] \begin{array}{rcl}dy_1 /dt &=& y_2 \\
dy_2/dt &=& -3 y_1 + 2 y_3 \\
dy_3/dt &=& y_4\\
dy_4/dt &=& y_1 - 2 y_3\end{array}
[/tex]
This can be written as
[tex] \frac{d}{dt} \pmatrix{y_1\\y_2\\y_3\\y_4}=
\pmatrix{0&1&0&0\\-3&0&2&0\\0&0&0&1\\1 &0 -2&0} \pmatrix{y_1\\y_2\\y_3\\y_4} [/tex]
This is of the form ##dY/dt = A Y##, where ##Y## is the column vector of ##y_i##s and ##A## is the ##4 \times 4## matrix above. The solution is of the form
[tex] Y(t) = \exp(A t) Y(0) [/tex]
You can compute the matrix exponential from the eigenvalues and eigenvectors (at least, if none of the eigenvalues are repeated). Google 'matrix exponential'.
so I got the Eigen values as i, -i, 2i, -2i

for i, i found an Eigen value of
-i
1
-i
1
Does this seem correct?
Now if i find the eigenvectors for the other eigenvalues. i can use this in the exponential equation and eventually reach values of x1 and x2 which should be the same answers from the previous part of the question?
 
  • #34
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I am also having trouble finding good sites online that delve into complex eigenvalues and eigenvectors for any matrix above 2x2. My textbook does touch on this and i was just wondering if anyone knew of anywhere good that i can find some examples to refer to.
 
  • #35
vela
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so I got the eigenvalues as i, -i, 2i, -2i

for i, i found an eigenvector of
-i
1
-i
1
Does this seem correct?
It's easy enough to check yourself. Multiply it into the matrix and see if you get ##i## times the vector back.
Now if i find the eigenvectors for the other eigenvalues. i can use this in the exponential equation and eventually reach values of x1 and x2 which should be the same answers from the previous part of the question?
Yes.
 
  • #36
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0
So now that I have my eigenvalues and eigenvectors. I believe I now use them for the exponential matrix and that should give me my sin and cos equations of x1 and x2 from earlier in the question. However I am unsure how I am able to reach this point as I have not fully learnt this yet and am unable to find suitable sources. Does anyone know of a decent source?
 
  • #37
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Are you familiar with ##e^{iθ}=cosθ+isinθ##? If the eigenvalues and eigenvectors were real, would you know how to proceed next?

Chet
 
  • #38
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ive only ever used it for 2x2 matrix's, I assume this way I would receive 4 equations with 4 unknowns? I'm still a little unsure as to how those unknowns are calculated, will the 4 equations be equal to the 4 y equations that Ray Vickson listed?
 
  • #39
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also I just want to double check, if I had the eigenvalue of 'i' and the corresponding eigenvector (-i,1,-i,1)
the first term of the first y equation would be -i(acost+ibsint)
with a and b as unknowns
and so on with each value and vector
 
Last edited:
  • #40
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0
Are you familiar with ##e^{iθ}=cosθ+isinθ##? If the eigenvalues and eigenvectors were real, would you know how to proceed next?

Chet
OK, so I believe I have gotten my four y equation out with 4 unknowns c1, c2, c3 and c4. What will these 4 equations equal so that I am able to determine the value of each 'c'
 
  • #41
Ray Vickson
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So now that I have my eigenvalues and eigenvectors. I believe I now use them for the exponential matrix and that should give me my sin and cos equations of x1 and x2 from earlier in the question. However I am unsure how I am able to reach this point as I have not fully learnt this yet and am unable to find suitable sources. Does anyone know of a decent source?
Check out the concept of "matrix function". Basically, if ##f(x) = c_0 + c_1 x + c_2 x^2 + \cdots## is an analytic function of ##x##, we can define ##f(A) = c_0 I + c_1 A + c_2 A^2 + \cdots## for an ##n \times n## matrix ##A##. Here, ##I = n \times n## unit matrix. It is a fact that for a matrix with ##n## distinct eigenvalues ##\lambda_1, \lambda_2, \ldots, \lambda_n## we can find ##n## matrices ##E_1, E_2, \ldots, E_n## such that
[tex] f(A) = E_1 f(\lambda_1) + E_2 f(\lambda_2) + \cdots + E_n f(\lambda_n) [/tex]
The crucial point is that the matrices ##E_1, E_2, \ldots, E_n## are independent of the function ##f##. That means we can find them by applying this formula successively to the functions ##f_0(x) = 1 = x^0, f_1(x) = x, f_2(x) = x^2, \ldots, f_{n-1}(x) = x^{n-1}## using the known (or computable) matrices ##I = A^0, A, A^2, \ldots, A^{n-1}##.

If you have repeated eigenvalues the formula becomes a bit more complicated.

See, eg., http://www.siam.org/books/ot104/OT104HighamChapter1.pdf or
Gantmacher, "Theory of Matrices, Vol. I", Chapter 5; now available free as http://www.maths.ed.ac.uk/~aar/papers/gantmacher1.pdf .
 
  • #42
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0
thanks for your help everyone, I've managed to get the correct answer using a 2x2 matrix made from the original equations
 

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