Answered: Illustrate the general system of n…

Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015

1st Edition

ISBN:9781680331141

Author:HOUGHTON MIFFLIN HARCOURT

Publisher:HOUGHTON MIFFLIN HARCOURT

Chapter5: Solving Systems Of Linear Equations

Section: Chapter Questions

Problem 13CT

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Question

Illustrate the general system of n first-order linear equations?

Expert Solution

Step 1

What is System of first order Linear Equation:

When there is more than one first order differential equations containing one independent variable and more than dependent variable, the system is referred to as a system of first order linear differential equation. Every higher order linear differential equation can be converted to a system of first order linear differential equations. The order of the parent equation determines the number of equations in the system.

To Discuss:

We discuss the general system of n first-order linear equations.

Step 2

A system of $n$ first order linear differential equation is represented by the equation,

$x' = A x + b (t) (1)$

Here, $A \in M_{n \times n} (F)$ and $x \in ℝ^{n}$ .

When $b (t) = 0$ , the system is referred to as a homogeneous system. When $b (t) \neq 0$ , the system is non-homogeneous.

For a homogeneous system, the general solution is only the complementary function which is the solution to the equation

$x' = A x$

The solution to the homogeneous system is

$x (t) = \sum_{i = 1}^{n} c_{i} v_{i} e^{λ_{i} t}$

Here, $c_{i}, i = 1, 2, 3, . . ., n$ are arbitrary constants, $λ_{i}, i = 1, 2, 3, . . ., n$ are the eigenvalues corresponding to the matrix $A$ and $v_{i}, i = 1, 2, 3, . ., n$ are the respective eigenvectors.

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