anonymous
  • anonymous
Solve the initial problem \[y_1''=2y_1+y_2+y_1'+y_2'\]\[y_2''=-5y_1+2y_2+5y_1'-y_2'\]\(y_1(0)=y_2(0)=y_1'(0)=4\), \(y_2'(0)=-4\)
Linear Algebra
schrodinger
  • schrodinger
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hba
  • hba
Integrate.
anonymous
  • anonymous
Note that I put this under linear algebra.
anonymous
  • anonymous
I think it has something to do with eigenvalues and eigenvectors

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anonymous
  • anonymous
and diagonalization too
experimentX
  • experimentX
let y'1 = u, and y'2 = v, you get 4x4 system.
anonymous
  • anonymous
Hmm.. How??
experimentX
  • experimentX
|dw:1357743467559:dw|
experimentX
  • experimentX
|dw:1357743488688:dw|
experimentX
  • experimentX
let that matrix be A, you get X' = AX <-- this is logistic equation ... I must admit ... to me, this is not a nice question.
experimentX
  • experimentX
the solution is \[ X = Se^{\Lambda t}S^{-1}X(0)\]
experimentX
  • experimentX
|dw:1357745773804:dw|
sirm3d
  • sirm3d
how about laplace transform?
experimentX
  • experimentX
|dw:1357746034677:dw||dw:1357746105903:dw|
experimentX
  • experimentX
|dw:1357746162435:dw|
experimentX
  • experimentX
|dw:1357746200518:dw||dw:1357746235140:dw|
experimentX
  • experimentX
so the final solution is |dw:1357746290760:dw|
anonymous
  • anonymous
@sirm3d Using the laplace transform is not as pretty as @experimentX 's cleaner linear algebra solution. You're not guaranteed invertible functions in the frequency domain.

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