FIRST ORDER SYSTEMS INTRODUCTION
LINEAR ALGEBRA REVIEW and THEORY

First Order System Problem General Form:

solve

$${\bf x}' = \left[\begin{array}{c}x'_1\\ x'_2\\ \vdots\\ x'_n\... ...in{array}{c}x_1^0\\ x_2^0\\ \vdots\\ x_n^0\end{array}\right] .$$

• If $F_1, \ldots, F_n$ and $\partial F_1/\partial x_1,\ \partial F_1/\partial x_2,\ \ldots,\ \partial F_n/\partial x_n$ are continuous in a region R defined by $\alpha<t<\beta$, $\alpha_1<x_1<\beta_1, \ldots, \alpha_n<x_n<\beta_n$, with $(t_0, x_1^0, \ldots x_n^0)$ in R, then there is an interval $t\in (t_0-h,t_0+h)$ for which there exists a unique solution to the first order system that satisfies the initial conditions.
• If the F_i's are linear functions of the x_i's, the system is linear; otherwise the system is nonlinear.
• A linear first order system has the form
  
  $${\bf x}'= \left[\begin{array}{c} p_{11}(t)x_1+\cdots+p_{1n}(t... ...+p_{nn}(t)x_n + g_n(t) \end{array}\right] = P{\bf x}+ {\bf g}.$$
  
  
  If all g_i's are zero, the system in homogeneous;
  otherwise the system is nonhomogeneous.
  If all p_ij's and all g_i's are continuous for $t\in I=(\alpha,\beta)$, and $t_0\in I$, then for all $t\in I$ there is a unique solution that satisfies the intitial conditions.

Conversion of n^th Order IVP to 1^st Order System

• Assume you are given the equation
  
  $$y^{(n)} = f(t,y, y', \ldots, y^{(n-1)}),$$
  
  
  with $y(t_0) = y_0,\ y'(t_0) = y'_0,\ \ldots,\ y^{(n-1)}(t_0) = y^{(n-1)}_0$.
• Let $x_1 = y,\ x_2 = y',\ \ldots,\ x_n = y^{(n-1)};$
  then the first order system is
  
  $${\bf x}'= \left[\begin{array}{c}x_2\\ x_3\\ \vdots\\ x_n\\ f(... ...{array}{l}y_0\\ y'_0\\ \vdots\\ y^{(n-1)}_0\end{array}\right] .$$
  
  

Basic Linear Algebra Review

• Vectors, Matrices, Vector Algebra and Matrix Algebra.
• Systems of Linear Equations $A{\bf x}= {\bf b}$.
• Linear Independence of Vectors.
• Eigenvalues $\lambda$ and Eigenvectors ${\bf x}$ satisfy $A{\bf x}=\lambda {\bf x}$:
  • Hermitian matrices satisfy A^* = A, where A^* is conjugate transpose of A; A^* = A^T if A is real.
    An $n \times n$ Hermitian A has n real eigenvalues and n linearly independent orthogonal eigenvectors.
  • An $n \times n$ diagonalizable matrix has n linearly independent eigenvectors. Let T be a matrix with the eigenvectors of Aas columns, and let D be the diagonal matrix with the respective eigenvalues as diagonal entries; then AT = TD and A = TDT^-1.
    If A is Hermitian, then T^-1 = T^*; T is orthogonal.

First Order Systems Basic Theory

Assume that we are given the first order system

$${\bf x}' = P(t){\bf x}+ {\bf g}(t) ,$$

where P is an $n \times n$ matrix and ${\bf x}$ is an $n \times 1$ vector.

• Superposition of Solutions
  Let ${\bf x}^{(i)} = [x_{1i}(t),\ x_{2i}(t),\ \ldots,\ x_{ni}(t)]^T$.
  If ${\bf x}^{(1)},\ {\bf x}^{(2)},\ \ldots,\ {\bf x}^{(k)}$ are solutions to ${\bf x}'= P{\bf x}$,
  then ${\bf x}= c_1{\bf x}^{(1)} + c_2{\bf x}^{(2)} + \cdots + c_k{\bf x}^{(k)}$ is a solution.
• The Wronskian: let $X(t)=[{\bf x}^{(1)}\ {\bf x}^{(2)}\ \cdots\ {\bf x}^{(n)}]$.
  The Wronksian $W[{\bf x}^{(1)},\ {\bf x}^{(2)},\ \ldots,\ {\bf x}^{(n)}] = \det(X).$
  • If ${\bf x}^{(1)},\ {\bf x}^{(2)},\ \ldots,\ {\bf x}^{(n)}$ are linearly independent
    solutions to ${\bf x}'= P{\bf x}$ for each $t \in (\alpha, \beta)$,
    then any solution can be expressed as a linear
    combination of ${\bf x}^{(1)},\ {\bf x}^{(2)},\ \ldots,\ {\bf x}^{(n)}$ in exactly one way.
  • A fundamental set, $\{{\bf x}^{(1)},\ {\bf x}^{(2)},\ \ldots,\ {\bf x}^{(n)}\}$, is any set of solutions to ${\bf x}'= P{\bf x}$ which is linearly independent for all $t \in (\alpha, \beta)$.
  • If ${\bf x}^{(1)},\ {\bf x}^{(2)},\ \ldots,\ {\bf x}^{(n)}$ are solutions to ${\bf x}'= P{\bf x}$ for $t \in (\alpha, \beta)$, then the Wronskian $W[{\bf x}^{(1)},\ {\bf x}^{(2)},\ \ldots,\ {\bf x}^{(n)}]$ either is always zero or is never zero for $t \in (\alpha, \beta)$.