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System of Differential Equations Help?
X' = (1 2 ) X
(-5 -1)
That is the equation (please forgive the bad notation, it is suppose to be the X' vector equals the matrix A times vector X). I got as far as calculating the two eigenvalues and eigenvectors which I verified were correct with the book but I always come out with a different final answer.
I would very much is someone could explain the entire process STEP by STEP all the way to the final answer in as much detail as possible. Thanks so much!
1 Answer
- 7 years agoFavorite Answer
given a system of linear first order differential equations,
1) Find the eigenvalues of the matrix
2) Find the associated eigenvalues
If the matrix is nxn, then you will have n solutions
Each solution Ya = e^(eigenvalue(a)*t)*(eigenspace(a))
The general solution is therefore Y = sum from [a=1 to n] of (Ca*e^(eigenvalue(a)*t)*(eigenspace(a)))
If you are given initial values, then you can find the all of the constant Cn values.