Problem 1 In Problems $1-6$, write the l... [FREE SOLUTION]

Chapter 10: Problem 1

In Problems $1-6$, write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=3 x-5 y \\ &\frac{d y}{d t}=4 x+8 y \end{aligned} $$

Short Answer

Expert verified

The matrix form is $ \frac{d\vec{X}}{dt} = \begin{bmatrix} 3 & -5 \\ 4 & 8 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} $.

Step by step solution

Identify the Coefficients

Examine the given system of differential equations: $ \frac{dx}{dt} = 3x - 5y $ and $ \frac{dy}{dt} = 4x + 8y $. The coefficients here are used for writing the matrix: 3 and -5 for the first equation, and 4 and 8 for the second equation.

Write the Coefficient Matrix

Place the coefficients of $ x $ and $ y $ from each equation into a matrix. This forms the coefficient matrix $ A $: \[A = \begin{bmatrix} 3 & -5 \ 4 & 8 \end{bmatrix}.\]

Define the Variables Vector

The variables $ x $ and $ y $ form a column vector represented as $\vec{X} = \begin{bmatrix} x \ y \end{bmatrix}.$

Write the Derivative Vector

The derivatives of $ x $ and $ y $ with respect to $ t $ are put into a vector: \[ \frac{d\vec{X}}{dt} = \begin{bmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{bmatrix}. \]

Combine into Matrix Form

Combine the coefficient matrix, the variables vector, and the derivative vector to write the system in matrix form: \[ \frac{d\vec{X}}{dt} = A\vec{X}, \]where \[ \frac{d\vec{X}}{dt} = \begin{bmatrix} \frac{dx}{dt} \ \frac{dy}{dt} \end{bmatrix}, \quad A = \begin{bmatrix} 3 & -5 \ 4 & 8 \end{bmatrix}, \quad \vec{X} = \begin{bmatrix} x \ y \end{bmatrix}. \]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Form

In linear algebra, expressing a system of equations in matrix form is both powerful and efficient. Essentially, matrix form is a way to represent a system of equations using matrices. Let's explore how it simplifies complex equations:

Matrix Representation: Instead of handling multiple separate equations, you can combine them into one concise format.
Organized Structure: Every element in a matrix represents a specific part of your equation, bringing order to potentially chaotic data.
Facilitates Solutions: By using advanced linear algebra techniques, solving becomes more straightforward.

In the context of differential equations, moving to matrix form is the first step in applying methods like eigenvalue-eigenvector analysis, which is essential for finding solutions. Thus, understanding how to convert a system into matrix form is crucial for both clarity and solving these equations effectively.

System of Differential Equations

A system of differential equations involves multiple equations that are related and need to be solved together. These systems appear often in real-world applications:

Dynamic Systems: They describe how variables change over time, such as in physics, engineering, and biology.
Interconnected Variables: Each equation in the system can depend on multiple variables, demonstrating intricacies within systems.
Unified Approach: Solving these involves using all the equations at once rather than separately.

Consider the example system given:- The rate of change of the variable $ x $ depends not only on $ x $ itself but also on $ y $, and vice versa.- This interdependence requires developing techniques that can solve these coupled equations simultaneously.Systems of differential equations often result in insights into how each variable influences the others, providing a complete picture of the modeled phenomena.

Coefficient Matrix

The coefficient matrix is a compact way to store the coefficients from the system of equations. Think of it as a summary that鈥檚 easy to use when manipulating equations algebraically:

Matrix Elements: Each number in the matrix corresponds to a coefficient from the equations, aligning with each variable.
Matrix Size: The dimension of the coefficient matrix corresponds to the number of equations and variables in the system.
Simplified Calculations: Having all coefficients in one place makes it easier to perform linear algebra operations.

For the example given, the coefficient matrix is \[A = \begin{bmatrix} 3 & -5 \ 4 & 8 \end{bmatrix}\]- Each row of the matrix represents an equation.- Each column matches a variable ($x$ or $y$).Utilizing the coefficient matrix in linear algebra allows for various manipulations and solutions methods such as finding determinants, calculating inverses, and performing Gaussian elimination. Ultimately, it is a vital tool for systematically and efficiently handling systems of equations.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Problems \(1-6\), write the linear system in matrix form. $$ \begin{aligned} &\frac{d x}{d t}=3 x-5 y \\ &\frac{d y}{d t}=4 x+8 y \end{aligned} $$

Short Answer

Step by step solution

Identify the Coefficients

Write the Coefficient Matrix

Define the Variables Vector

Write the Derivative Vector

Combine into Matrix Form

Key Concepts

Matrix Form

System of Differential Equations

Coefficient Matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Electricity

Magnetism

Magnetism and Electromagnetic Induction

Radiation

Engineering Physics

Study anywhere. Anytime. Across all devices.