Problem 2 - Differential Equations and Linear Algebra

Question

Solve the given system of differential equations. $$x_{1}^{\prime}=2 x_{1}-3 x_{2}, \quad x_{2}^{\prime}=x_{1}-2 x_{2}$$

Step-by-Step Solution

Answer

The general solution to the given system of differential equations is: \[ X(t) = c_1 e^{\frac{1 + \sqrt{3}i}{2}t} \begin{pmatrix} 3 \\ 1 + \sqrt{3}i \end{pmatrix} + c_2 e^{\frac{1 - \sqrt{3}i}{2}t} \begin{pmatrix} 3 \\ 1 - \sqrt{3}i \end{pmatrix} \] where \(c_1\) and \(c_2\) are constants.

Step 1: Rewrite the system in matrix form

We have the system of differential equations: \[ x_1' = 2x_1 - 3x_2 \] \[ x_2' = x_1 - 2x_2 \] Rewrite this system in matrix form: \[ \begin{pmatrix} x_1' \\ x_2' \end{pmatrix} = \begin{pmatrix} 2 & -3 \\ 1 & -2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \] Which can be denoted as: \[ X' = AX \] Where: \[ X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \, , A = \begin{pmatrix} 2 & -3 \\ 1 & -2 \end{pmatrix} \]

Step 2: Find the eigenvalues and eigenvectors

To find the eigenvalues, we need to solve the following equation: \[ \det(A - \lambda I) = 0 \] Where \(I\) is the identity matrix and \(\lambda\) is the eigenvalue. So we have: \[ \det \begin{pmatrix} 2 - \lambda & -3 \\ 1 & -2 - \lambda \end{pmatrix} = (2-\lambda)((-2)-\lambda) - (-3)(1) = 0 \] Solve for \(\lambda\): \[ \lambda^2 - 1\lambda + 1 = 0 \] The two eigenvalues are \( \lambda_1 = \frac{1 + \sqrt{3}i}{2} \) and \( \lambda_2 = \frac{1 - \sqrt{3}i}{2} \). Next, we will find the eigenvectors associated with these eigenvalues. For \(\lambda_1\): \[ (A - \lambda_1 I) V_1 = 0 \] \[ \begin{pmatrix} 2 - \lambda_1 & -3 \\ 1 & -2 - \lambda_1 \end{pmatrix} \begin{pmatrix} v_{11} \\ v_{21} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] Solve for \(V_1\), we get: \[ V_1 = \begin{pmatrix} 3 \\ 1 + \sqrt{3}i \end{pmatrix} \] Similarly, for \(\lambda_2\): \[ (A - \lambda_2 I) V_2 = 0 \] \[ \begin{pmatrix} 2 - \lambda_2 & -3 \\ 1 & -2 - \lambda_2 \end{pmatrix} \begin{pmatrix} v_{12} \\ v_{22} \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} \] Solve for \(V_2\), we get: \[ V_2 = \begin{pmatrix} 3 \\ 1 - \sqrt{3}i \end{pmatrix} \]

Step 3: Construct the general solution

Now we have the eigenvalues and the eigenvectors, we can construct the general solution as follows: \[ X(t) = c_1 e^{\lambda_1t} V_1 + c_2 e^{\lambda_2t} V_2 \] \[ X(t) = c_1 e^{\frac{1 + \sqrt{3}i}{2}t} \begin{pmatrix} 3 \\ 1 + \sqrt{3}i \end{pmatrix} + c_2 e^{\frac{1 - \sqrt{3}i}{2}t} \begin{pmatrix} 3 \\ 1 - \sqrt{3}i \end{pmatrix} \] Where \(c_1\) and \(c_2\) are constants. This is the general solution to the given system of differential equations.

Key Concepts

System of Differential Equations

A system of differential equations consists of multiple differential equations that involve multiple dependent variables and their derivatives. These systems are crucial in modeling real-world phenomena where several variables interact simultaneously, such as in engineering, physics, and biology. For instance, the system considered here is:

• \( x_1' = 2x_1 - 3x_2 \)
• \( x_2' = x_1 - 2x_2 \)

This example showcases two first-order linear differential equations involving variables \( x_1 \) and \( x_2 \) with their respective derivatives \( x_1' \) and \( x_2' \).
This type of system is often solved using methods like elimination, substitution, or, as in this example, matrix methods. The goal is to find functions \( x_1(t) \) and \( x_2(t) \) that satisfy both equations simultaneously across time \( t \). Analyzing such systems helps us understand how components within a system interact over time.

Matrix Form of Differential Equations

Matrix representation is a powerful method for dealing with systems of differential equations. Converting a system into matrix form unifies the system into a concise, efficient format. By using matrices and vectors, it becomes easier to apply linear algebra techniques. For the given system of equations:

• \( x_1' = 2x_1 - 3x_2 \)
• \( x_2' = x_1 - 2x_2 \)

We rewrite it in matrix form:\[\begin{pmatrix} x_1' \ x_2' \end{pmatrix} = \begin{pmatrix} 2 & -3 \ 1 & -2 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix}\]where the matrix \( A = \begin{pmatrix} 2 & -3 \ 1 & -2 \end{pmatrix} \) contains the coefficients of the system. This matrix form \( X' = AX \) allows us to utilize matrix operations to find the solution. This approach is advantageous as it simplifies the process and extends naturally to systems with more equations and variables.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental in solving systems of linear differential equations in matrix form. These concepts stem from linear algebra and help describe important properties of matrix transformations. For the matrix \( A \) derived from our system:\[A = \begin{pmatrix} 2 & -3 \ 1 & -2 \end{pmatrix}\]We find the eigenvalues \( \lambda \) by solving the characteristic equation \( \det(A - \lambda I) = 0 \). Solving this, we get the eigenvalues:

• \( \lambda_1 = \frac{1 + \sqrt{3}i}{2} \)
• \( \lambda_2 = \frac{1 - \sqrt{3}i}{2} \)

For each eigenvalue, we determine its eigenvector by substituting \( \lambda \) back into the equation \( (A - \lambda I)V = 0 \). For our solution, respective eigenvectors for \( \lambda_1 \) and \( \lambda_2 \) are:

• \( V_1 = \begin{pmatrix} 3 \ 1 + \sqrt{3}i \end{pmatrix} \)
• \( V_2 = \begin{pmatrix} 3 \ 1 - \sqrt{3}i \end{pmatrix} \)

This process helps us construct the general solution of our system.

General Solution of Differential Equations

The general solution of a system of differential equations involving eigenvalues and eigenvectors encapsulates all possible solutions of the system. In our specific example, once we have both eigenvalues and their corresponding eigenvectors, we can express the solution as a combination of these elements:

• \( X(t) = c_1 e^{\lambda_1 t} V_1 + c_2 e^{\lambda_2 t} V_2 \)

This translates to:\[X(t) = c_1 e^{\frac{1 + \sqrt{3}i}{2}t} \begin{pmatrix} 3 \ 1 + \sqrt{3}i \end{pmatrix} + c_2 e^{\frac{1 - \sqrt{3}i}{2}t} \begin{pmatrix} 3 \ 1 - \sqrt{3}i \end{pmatrix}\]Here, \( c_1 \) and \( c_2 \) are constants determined by the initial conditions of the system. These solutions are often expressed in terms of exponential functions involving complex numbers, which can be interpreted and transformed to fit real-world scenarios by using techniques such as Euler's formula. This solution completely describes the behavior of the system across time.