 # Chapter Review Sheets for Elementary Differential Equations and Boundary Value Problems

## Chapter 7: Systems of First Order Linear Equations

Definitions:
•Systems of ODE's
Linear vs . Nonlinear Systems
•Solution
•Homogenous and Nonhomogeneous Systems Matrix ,
•(Inner) Product , Orthogonal
•Nonsingular (Invertible) and Singular (Noninvertible)
Row Reduction ( Gaussian Elimination )
Linear Systems , Homogeneous, Nonhomogeneous
•Augmented Matrix
Linear Dependence and Independence
•Eigenvalues, Eigenvectors, Generalized Eigenvectors
•Normalization
Multiplicity m , Simple Multiplicity (m = 1)
•General Solution, Fundamental Set of Solutions
•Phase Plane, Phase Portrait
•Generalized Eigenvector
•Node, Saddle Point, Spiral Point, Improper Node
•Fundamental Matrix
•The matrix exp(At)
•Similarity Transformation, Diagonalizable Matrices

Theorems:
•Theorem 7.1. l: Existence and uniqueness of solutions for general systems of First Order IVP's
(p. 358)
•Theorem 7.1.2. Existence and uniqueness of solutions for linear systems (p. 359)
•Theorem 7.4.1: Superposition of solutions (p. 386)
•Theorem 7.4.2: Theorem Superposition of solutions – general case (p. 387)
•Theorem 7.4.3: Nonvanishing of Wronskian for linearly independent solutions (p. 387)
•Theorem 7.4.4: Existence of fundamental set of solutions (p. 388)

Important Skills:
•Representation of solutions and vectors
•Find the inverse of a matrix. (Ex. 2, p. 369)
•Find the solution to a set of linear algebraic equations . (Ex. 1, p. 375)
•Determine if a set of vectors is linearly independent. (Ex. 3, p. 377)
•Find the eigenvalues and eigenvectors of a matrix. (Ex. 5, p. 381)
•Sketch a direction field for a 2 x 2 system of linear ODE's. (Ex. 2, p. 394)
•Find the general solution of a system of linear ODE's.
•Distinct Eigenvalues (Ex. 3, p. 397)
Complex Eigenvalues (Ex. 1, p. 401)
•Repeated Eigenvalues (Ex. 2, p. 423)
•Find the fundamental matrix for a system of linear ODE's. (Ex. 1 & 2, p. 414-415)
•Find the similarity transformation to diagonalize a matrix. (Ex. 3, p. 418)
•Use the method of undetermined coefficients to find the particular solution to a nonhomogeneous
linear system of ODE's. (Ex. 2, p. 435)
•Use the method of variation of parameters to find the particular solution to a nonhomogeneous
linear system of ODE's. (Ex. 3, p. 437)

Relevant Applications:
•Multiple Spring Mass Problems, Multiple Tank Mixture Problems

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