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

# 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 ,

•Transpose, Conjugate, Adjoint, Determinant Scalar

•(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)

•Self Adjoint (Hermitian)

•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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