Question

Find the general solution of each of the linear systems in Exercises 1-26. $$ \begin{aligned} &x^{\prime}=8 x-4 y \\ &y^{\prime}=x+4 y \end{aligned} $$

Short answer

The general solution of the given linear system is: \[ x(t) = c_1 e^{8t} \begin{pmatrix} 1 \\ 0 \end{pmatrix} + c_2 e^{4t} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \] Where \(c_1\) and \(c_2\) are constants determined by the initial conditions.

Step-by-step solution

Rewrite as a matrix equation

The given linear system can be written as a matrix equation as follows: \[ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 8 & -4 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \]

Find eigenvalues

To find the eigenvalues, we need to solve the characteristic equation, which is given by: \[ \text{det} (A - \lambda I) = 0 \] Where A is the matrix, \(\lambda\) are the eigenvalues, and I is the identity matrix. In our case: \[ \text{det} \begin{pmatrix} 8-\lambda & -4 \\ 1 & 4-\lambda \end{pmatrix} = (8-\lambda)(4-\lambda) - (-4)(1) = 0 \] Solving for \(\lambda\), we get: \( \lambda^2 - 12\lambda + 32 = 0 \) By factoring the quadratic equation, we get: \( (\lambda-8)(\lambda - 4) = 0 \) So, the eigenvalues are \(\lambda_1 = 8\) and \(\lambda_2 = 4\).

Find eigenvectors

Next, we need to find the eigenvectors corresponding to the eigenvalues. For \(\lambda_1 = 8\): We need to solve the linear system formed by the equation: \( (A - 8I) v_1 = 0 \) \[ \begin{pmatrix} (8-8) & -4 \\ 1 & (4-8) \end{pmatrix} \begin{pmatrix} v_{1x} \\ v_{1y} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] This simplifies to: \[ \begin{pmatrix} 0 & -4 \\ 1 & -4 \end{pmatrix} \begin{pmatrix} v_{1x} \\ v_{1y} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] One possible eigenvector for \(\lambda_1 = 8\) is \(v_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\). For \(\lambda_2 = 4\): We need to solve the linear system formed by the equation: \( (A - 4I) v_2 = 0 \) \[ \begin{pmatrix} (8-4) & -4 \\ 1 & (4-4) \end{pmatrix} \begin{pmatrix} v_{2x} \\ v_{2y} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] This simplifies to: \[ \begin{pmatrix} 4 & -4 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} v_{2x} \\ v_{2y} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] One possible eigenvector for \(\lambda_2 = 4\) is \(v_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}\).

Form general solution

The general solution of the given linear system is the linear combination of the eigenvectors multiplied by their respective \(e^{\lambda t}\) terms: \[ x(t) = c_1 e^{8t} \begin{pmatrix} 1 \\ 0 \end{pmatrix} + c_2 e^{4t} \begin{pmatrix} 1 \\ 1 \end{pmatrix} \] Where \(c_1\) and \(c_2\) are constants determined by the initial conditions.

Key concepts

General Solution

A general solution in the context of systems of linear differential equations refers to the combination of all possible solutions that satisfy the system. In our problem, the system is represented by differential equations involving functions and their derivatives. To find the general solution, we utilize the eigenvectors associated with the eigenvalues derived from the coefficient matrix. This involves several key steps:

• First, we translate the system of equations into a matrix form, which simplistically packages the interactions between variables.


• Next, we identify eigenvalues and eigenvectors. These are fundamental because the solution of a system of linear differential equations essentially revolves around these components.


• Finally, we express the general solution as a linear combination of terms involving these eigenvectors and exponential functions of their corresponding eigenvalues.

This process provides a comprehensive depiction of all possible behavior the system can depict over time. The constants in the general solution are determined from initial conditions, specifying the exact path within this family of solutions.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are the backbone of solving linear differential equations and understanding the dynamics of a system. An eigenvalue, \( \lambda \), represents the factor by which an associated eigenvector is scaled during a transformation described by the matrix. Let’s break down their importance:

• **Eigenvalues** are numbers obtained by solving the characteristic equation, derived from the matrix by subtracting \( \lambda I \) from it and setting the determinant to zero. They tell us about the stability and behavior of the system.


• **Eigenvectors** are the "directions" that remain unchanged (only scaled) when a matrix acts upon them. These provide insight into the nature of the system's response.

By analyzing eigenvalues:

• Positive eigenvalues imply growth, indicating that solutions will grow exponentially.
• Negative eigenvalues imply decay, indicating that solutions will shrink over time.

Eigenvectors, on the other hand, reveal the principal directions in which this growth or decay occurs, forming the essential building blocks of the solution's structure.

Systems of Differential Equations

Systems of differential equations involve multiple interrelated differential equations that describe various dynamic systems. They frequently occur in fields like physics, engineering, and biology. Understanding these systems entails converting them into a solvable matrix form, as was done in our exercise. Here's how they come into play:

• Such systems often describe real-world phenomena where multiple quantities interact and change over time, like population dynamics or electrical circuits.


• By representing the system as a matrix equation, it becomes easier to manipulate algebraically and set up for methods of solution like finding eigenvalues and eigenvectors.


• The solution of these systems can reveal steady states, cycles, or other dynamic behaviors central to the modeling of complex phenomena.

Using a structured approach to solve these systems helps in predicting behaviors accurately. Despite the potentially complex interrelations, organizing the system neatly as matrices and using eigen-based approaches provide insightful and manageable solutions.