The Characteristic Polynomial of a Linear Homogeneous nth Order ODE with Constant Coefficients Theorem 1: Let \(y^{(n)} + a_{n-1}y^{(n-1)} + ... + a_1y' + a_0y = 0\) be a linear homogeneous $n^{\mathrm{th}}$ order ODE with constant coefficients. Then the characteristic polynomial $p(\lambda) = \lambda^n + a_{n-1}\lambda^{n-1} + ... + a_1\lambda + a_0$ is the same as the … [Read more...]

## The Wronskian of a Linear Homogeneous nth Order ODE

FoldUnfold Table of Contents The Wronskian of a Linear Homogeneous nth Order ODE The Wronskian of a Linear Homogeneous nth Order ODE Recall from the Fundamental Sets and Matrices of a Linear Homogeneous nth Order ODE page that if we have a linear homogeneous $n^{\mathrm{th}}$ order ODE $y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0y = 0$ then a linearly independent … [Read more...]

## Fundamental Sets and Matrices of a Linear Homogeneous nth Order ODE

FoldUnfold Table of Contents Fundamental Sets and Matrices of a Linear Homogeneous nth Order ODE Fundamental Sets and Matrices of a Linear Homogeneous nth Order ODE Consider a linear homogeneous $n^{\mathrm{th}}$ order ODE: (1) \begin{align} \quad y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + a_1(t)y' + a_0(t)y = 0 \end{align} We can define what it means for a collection of … [Read more...]

## The Companion Matrix of a Linear Homogeneous nth Order ODE

FoldUnfold Table of Contents The Companion Matrix of a Linear Homogeneous nth Order ODE The Companion Matrix of a Linear Homogeneous nth Order ODE We will now discuss some of the theory regarding linear homogeneous nth order ODEs. Recall that a linear homogeneous $n^{\mathrm{th}}$ order ODE can be written as: (1) \begin{align} \quad y^{(n)} + a_{n-1}(t)y^{(n-1)} + ... + … [Read more...]

## The Unique Solutions to Linear Nonhomogeneous Systems of First Order ODEs

FoldUnfold Table of Contents The Unique Solutions to Linear Nonhomogeneous Systems of First Order ODEs The Unique Solutions to Linear Nonhomogeneous Systems of First Order ODEs Theorem 1: Let $\mathbf{x}' = A(t)\mathbf{x} + g(t)$ with $\mathbf{x}(\tau) = \xi$ be a linear nonhomogeneous system of first order ODEs where $(\tau, \xi) \in D$, and let $\Phi (t, \tau)$ be the … [Read more...]

## Basic Properties of the State Transition Matrix to a Linear Homogeneous System of First Order ODEs

FoldUnfold Table of Contents Basic Properties of the State Transition Matrix to a Linear Homogeneous System of First Order ODEs Basic Properties of the State Transition Matrix to a Linear Homogeneous System of First Order ODEs Recall from The State Transition Matrix to a Linear Homogeneous System of First Order ODEs page that the state transition matrix to a linear … [Read more...]

## The State Transition Matrix to a Linear Homogeneous System of First Order ODEs

FoldUnfold Table of Contents The State Transition Matrix to a Linear Homogeneous System of First Order ODEs The State Transition Matrix to a Linear Homogeneous System of First Order ODEs We have already defined what a fundamental matrix to a linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ is. We noted that these fundamental matrices are not … [Read more...]

## The Change of Basis Theorems for Fundamental Matrices of a Linear Homogeneous System of First Order ODEs

FoldUnfold Table of Contents The Change of Basis Theorems for Fundamental Matrices of a Linear Homogeneous System of First Order ODEs The Change of Basis Theorems for Fundamental Matrices of a Linear Homogeneous System of First Order ODEs Theorem 1: If $\Phi$ is a fundamental matrix to the linear homogeneous system of first order ODEs $\mathbf{x}' = A(t)\mathbf{x}$ on … [Read more...]

## Criterion for a Matrix to be a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs

FoldUnfold Table of Contents Criterion for a Matrix to be a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs Criterion for a Matrix to be a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs Theorem 1: Let $\Phi$ be a solution to the matrix equation $X' = A(t)X$ on $J = (a, b)$. Then $\Phi$ is a fundamental matrix of the linear … [Read more...]

## Basic Properties of a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs

FoldUnfold Table of Contents Basic Properties of a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs Basic Properties of a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs Recall from The Determinant of a Fundamental Matrix to a Linear Homogeneous System of First Order ODEs page that if $\Phi$ is a solution to the matrix equation … [Read more...]