Systems of ODEs — Eigenvalue Methods

GRE Math Subject · Axiom Academy

LESSON Systems of ODEs -- Eigenvalue Methods Solving linear systems via eigenvalues, eigenvectors, and matrix exponentials A first-order linear system of ODEs with constant coefficients can be written in matrix form: where is an constant matrix and is a vector of unknown functions. For a system, this looks like: Key idea: Guess where is a scalar and is a constant vector. Substituting gives -- an eigenvalue problem. Solving the System Step by Step 2. For each eigenvalue , find eigenvector by solving 3. Each eigenvalue-eigenvector pair gives a solution This works perfectly when all eigenvalues are distinct. The three cases (distinct real, complex, repeated) each require slightly different treatment. Complex Eigenvalues Give Real Solutions When is real and has complex eigenvalues with eigenvectors , we extract two real solutions: Here . The solutions are pure rotation (no growth or decay): Defective Matrices and Generalized Eigenvectors If is a repeated eigenvalue but has only one linearly independent eigenvector (geometric multiplicity 1), we need a generalized eigenvector . Find such that (where is the eigenvector). Generalized eigenvector: solve The eigenvalues of determine the qualitative behavior near the origin: GRE shortcut: The origin is asymptotically stable iff all eigenvalues have negative real part . For : equivalent to and 0"> . Memorize: For a Jordan block : Any th-order ODE can be converted to a first-order system.

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