Systems of Differential Equations

Systems of Differential Equations

Matrix Form

Eigenvalue Method

Distinct Real Eigenvalues

If has distinct real eigenvalues, the solution is:

Complex Eigenvalues

If with eigenvector :

Repeated Eigenvalues

If has multiplicity with fewer than linearly independent eigenvectors, use generalized eigenvectors.

Matrix Exponential

Example

Eigenvalues: ,

Eigenvectors: ,

Solution:

Phase Plane Analysis

Non-Homogeneous Systems

Particular solution via variation of parameters:

Stability

Applications

Laplace Transform Method