Eigenvalue Method for Linear Systems of ODEs
For x' = Ax, each eigenpair (lambda, v) yields a solution e^(lambda t)v, and the general solution is their linear combination; complex eigenvalues produce spiraling sine-cosine solutions and defective eigenvalues require generalized eigenvectors. This reduces solving a coupled n-dimensional system to the algebra of the matrix A.
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For x' = Ax, each eigenpair (lambda, v) yields a solution e^(lambda t)v, and the general solution is their linear combination; complex eigenvalues produce spiraling sine-cosine solutions and defective eigenvalues require generalized eigenvectors. This reduces solving a coupled n-dimensional system to the algebra of the matrix A.
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