FoldUnfold Table of Contents The Characteristic Polynomial of a Linear Homogeneous nth Order ODE with Constant Coefficients 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 … [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...]