Fundamental matrix to solve IVP

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Mathismylove

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For the matrix [MATH] A = \begin{bmatrix} 1 & 3\\ 3& 1 \end{bmatrix} [/MATH]


I have the eigenvalues as -2 and 4, with eigenvectors

[MATH] \begin{bmatrix} -1\\ 1 \end{bmatrix}[/MATH] and [MATH]\begin{bmatrix} -1\\ 1 \end{bmatrix} [/MATH] through which I have obtained the matrix [MATH] \begin{bmatrix} e^{-2t} & e^{4t}\\ -e^{-2t} & e^{4t} \end{bmatrix} [/MATH]

Now to solve the IVP [MATH] y' = Ay [/MATH] with [MATH] y_0 = \begin{bmatrix} 2\\ 4 \end{bmatrix} [/MATH]


Can I simply plug in the matrix [MATH] P = \begin{bmatrix} e^{-2t} & e^{4t}\\ -e^{-2t} & e^{4t} \end{bmatrix} [/MATH] in place of [MATH]A[/MATH] and [MATH]y_0 [/MATH] in place of [MATH]y[/MATH] to solve the IVP?
If not what is the matrix [MATH]P[/MATH] called if it isn't the fundamental matrix (eventhough the determinant is non zero) and how to solve the IVP then.



I am really confused as to how to determine which matrix is the fundamental matrix and how to utilise it to solve the IVP (as different texts seems to be showing different information on this)

Please help
 
Mathismylove said:
For the matrix [MATH] A = \begin{bmatrix} 1 & 3\\ 3& 1 \end{bmatrix} [/MATH]


I have the eigenvalues as -2 and 4, with eigenvectors

[MATH] \begin{bmatrix} -1\\ 1 \end{bmatrix}[/MATH] and [MATH]\begin{bmatrix} -1\\ 1 \end{bmatrix} [/MATH] through which I have obtained the matrix [MATH] \begin{bmatrix} e^{-2t} & e^{4t}\\ -e^{-2t} & e^{4t} \end{bmatrix} [/MATH]

Now to solve the IVP [MATH] y' = Ay [/MATH] with [MATH] y_0 = \begin{bmatrix} 2\\ 4 \end{bmatrix} [/MATH]


Can I simply plug in the matrix [MATH] P = \begin{bmatrix} e^{-2t} & e^{4t}\\ -e^{-2t} & e^{4t} \end{bmatrix} [/MATH] in place of [MATH]A[/MATH] and [MATH]y_0 [/MATH] in place of [MATH]y[/MATH] to solve the IVP?
If not what is the matrix [MATH]P[/MATH] called if it isn't the fundamental matrix (even though the determinant is non zero) and how to solve the IVP then.



I am really confused as to how to determine which matrix is the fundamental matrix and how to utilize it to solve the IVP (as different texts seems to be showing different information on this)

Please help
Click to expand...
Please post the COMPLETE assignment EXACTLY as it was given to you.
 
Mathismylove said:
I

I am asked to find the fundamental matrix and then using that to find the unique solution of the IVP
Click to expand...
Post a photocopy of your assignment. That would tell us the COMPLETE assignment EXACTLY.
 
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