Problem 22 - Differential Equations and Linear Algebra

Question

Use the ideas introduced in this section to solve the given system of differential equations. $$x_{1}^{\prime}=x_{2}, \quad x_{2}^{\prime}=-x_{1}$$

Step-by-Step Solution

Answer

The solution for the given system of differential equations is: \(x(t) = c_1(\cos t + i \sin t) \begin{bmatrix} 1 \\ -i \end{bmatrix} + c_2 (\cos t - i \sin t) \begin{bmatrix} 1 \\ i \end{bmatrix}\) Which yields the system: \(x_1(t) = c_1(\cos t + i \sin t) + c_2(\cos t - i \sin t)\) \(x_2(t) = c_1 (-\sin t - i \cos t) + c_2 (-\sin t + i \cos t)\) The constants \(c_1\) and \(c_2\) depend on the initial conditions given.

Step 1: Setting up the system matrix A

The differential equations can be written as a matrix, thus: \( \begin{bmatrix} x_1' \\ x_2' \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \) Matrix A is: \( A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\)

Step 2: Compute eigenvalues and eigenvectors of A

Find the determinant of \(A - \lambda I\), which is the characteristic equation: \( det(A - \lambda I) = det \begin{bmatrix} - \lambda & 1 \\ -1 & -\lambda \end{bmatrix} = \lambda^2 + 1 \) Solve the characteristic equation for eigenvalues: \( \lambda^2 + 1 = 0 \implies \lambda = \pm i \) Next, find the eigenvectors for each eigenvalue: For \(\lambda = i\), \( (A - iI)v_1 = \begin{bmatrix} -i & 1 \\ -1 & -i \end{bmatrix} \begin{bmatrix} v_{1,1} \\ v_{1,2} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\) Choose an eigenvector \(v_1\): \( v_1 = \begin{bmatrix} 1 \\ -i \end{bmatrix} \) For \(\lambda = -i\), \( (A + iI)v_2 = \begin{bmatrix} i & 1 \\ -1 & i \end{bmatrix}\begin{bmatrix} v_{2,1} \\ v_{2,2} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\) Choose an eigenvector \(v_2\): \( v_2 = \begin{bmatrix} 1 \\ i \end{bmatrix} \)

Step 3: Applying the general solution

The general solution for this system is: \(x(t) = c_1 e^{\lambda_1 t} v_1 + c_2 e^{\lambda_2 t} v_2\) Plugging in the eigenvalues and eigenvectors: \(x(t) = c_1 e^{it} \begin{bmatrix} 1 \\ -i \end{bmatrix} + c_2 e^{-it} \begin{bmatrix} 1 \\ i \end{bmatrix}\) Using Euler's identity, this solution can be written in the sinusoidal form: \(x(t) = c_1(\cos t + i \sin t) \begin{bmatrix} 1 \\ -i \end{bmatrix} + c_2 (\cos t - i \sin t) \begin{bmatrix} 1 \\ i \end{bmatrix}\) Now, we obtain the system: \(x_1(t) = c_1(\cos t + i \sin t) + c_2(\cos t - i \sin t)\) \(x_2(t) = c_1 (-\sin t - i \cos t) + c_2 (-\sin t + i \cos t)\) The values of \(c_1\) and \(c_2\) will depend on the initial conditions provided, if any.

Key Concepts

System of Differential Equations

Understanding a system of differential equations can greatly simplify the process of analyzing and solving multiple related differential equations simultaneously. Let's consider a relatively simple system involving two variables, like the one in our example. We have two equations:
\[ x_{1}^{\prime}=x_{2} \]
\[ x_{2}^{\prime}=-x_{1} \]
Here, instead of dealing with each equation separately, we treat them as a system. This means we are looking for a set of functions, \( x_1(t) \) and \( x_2(t) \), that satisfy both equations at the same time.

• A system of differential equations describes how multiple quantities change relative to each other over time.
• These systems can be linear or nonlinear. Linear systems, like the example given, are easier to handle mathematically.
• The main goal is to find the general solution for the functions involved, which may require finding eigenvalues and eigenvectors as part of the process.

Eigenvalues and Eigenvectors

The key to solving many systems of differential equations lies in identifying the eigenvalues and eigenvectors of the system matrix. These mathematical concepts reveal important characteristics of the system's behavior over time.

• **Eigenvalues** (\( \lambda \)) indicate how the system evolves, such as growth rates or oscillation frequencies. They can be real or complex numbers.
• In our example, the characteristic equation derived from the system matrix yields two complex eigenvalues: \( \lambda = i \) and \( \lambda = -i \).
• **Eigenvectors** are vectors that remain in the same direction after transformation by the matrix, though their magnitude may change. They help form the building blocks for the solution of the system.

For the eigenvalue \( \lambda = i \), the corresponding eigenvector is \( \begin{bmatrix} 1 \ -i \end{bmatrix} \). For \( \lambda = -i \), it is \( \begin{bmatrix} 1 \ i \end{bmatrix} \).
Using these eigenvalues and eigenvectors, we construct the general solutions, which will describe the behavior of both functions \( x_1(t) \) and \( x_2(t) \) over time.

Matrix Representation

Matrix representation is a powerful tool in differential equations as it simplifies handling systems, especially when there are multiple equations at play. By using matrices, we can represent and calculate the interactions between each equation compactly and efficiently.
In the given system, the differential equations are converted into matrix form:
\[\begin{bmatrix} x_1' \ x_2' \end{bmatrix} = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\]

• This matrix equation succinctly captures the dynamics of the system, where matrix \( A = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \) encodes how the components affect each other.
• Matrices allow for straightforward computation of eigenvalues and eigenvectors, crucial for finding solutions.
• Matrix representation enables the use of advanced linear algebra techniques to analyze the system's behavior, including stability and oscillatory patterns.

Through matrix methods, not only do we manage the complexity of the system, but we also open the door to deeper insights into the relationships between variables.