Question

$\begin{aligned}-x_{1}+2 x_{2} &=0 \\ 2 x_{1}+3 x_{2} &=0 \end{aligned}$

   $\begin{aligned}-x_{1}+2 x_{2} &=0 \\ 2 x_{1}+3 x_{2} &=0 \end{aligned}$

Fundamentals of Differential Equations

Fundamentals of Differential Equations

R. Kent Nagle 9th Edition

Chapter 9, Problem 5

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Solved by Expert Ahmad Reda

08/29/2020

Step 1

This gives us: $$ \begin{aligned} -2x_{1}+4 x_{2} &=0 \\ 2 x_{1}+3 x_{2} &=0 \end{aligned} $$ Show more…

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$\begin{aligned}-x_{1}+2 x_{2} &=0 \\ 2 x_{1}+3 x_{2} &=0 \end{aligned}$

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Key Concepts

System of Linear Equations

This concept refers to a collection of two or more linear equations involving the same set of variables. Solving these systems involves finding values for the variables that satisfy all equations simultaneously. These systems form the foundation of many areas in algebra and are applicable in various fields, including engineering, economics, and the physical sciences.

Homogeneous Systems

A homogeneous system is a system of linear equations where all of the constant terms are zero. Such systems always have at least one solution, known as the trivial solution (where all variables are zero), and they are key to understanding concepts like linear independence and the structure of solution spaces in linear algebra.

Matrix Representation and Determinants

Representing systems of linear equations in matrix form (as AX = 0 for homogeneous systems) is a powerful method that leverages matrix operations and determinant properties. The determinant of the coefficient matrix can indicate whether a unique trivial solution exists (a nonzero determinant) or if there are infinitely many solutions (a zero determinant), linking the idea of invertibility to the solvability of the system.

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