Problem 19 Determine the eigenvalues for th... [FREE SOLUTION] (2024)

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Differential equations are equations that involve functions and their derivatives. They describe various phenomena in engineering, physics, economics, and other sciences. The exercise you encountered involves a system of linear differential equations. In matrix form, they can be represented as a vector equation, which simplifies solving them using linear algebra techniques. For example,

• The system can be expressed as \(X' = AX\), where \(X\) is the vector of variables and \(A\) is the coefficient matrix.
• This matrix form helps in using matrix algebra to find solutions.

The key to solving such systems lies in finding eigenvalues and eigenvectors of the matrix \(A\).

Matrix algebra involves operations with matrices, crucial for solving systems of linear equations. By converting differential equations into matrix form, you can leverage matrix operations to find solutions. The matrix representative worked here is: \[ \begin{pmatrix} x' \ y' \ \end{pmatrix} = \begin{pmatrix}-5 & 10 \ -4 & 7 \ \end{pmatrix} \begin{pmatrix} x \ y \ \end{pmatrix} \]

• Eigenvalues are found by solving the characteristic equation derived from \[(A - \lambda I)X = 0\], where \(I\) is the identity matrix.
• Algebras such as matrix multiplication, determinants, and eigenvalue calculations simplify understanding the behavior of the system.
  Hence, matrix algebra serves as a bridge to translate the complex behavior of systems into solvable algebraic problems.

Eigenvalues can be real or complex and crucial to understanding a system's behavior. When solving the characteristic equation, you might encounter complex eigenvalues. In this exercise, we find:
The characteristic equation is \[ \det \( \begin{pmatrix} -5 - \lambda & 10 \ -4 & 7 - \lambda \ \end{pmatrix} \) = 0 \]
Solving this gives:
\[ \lambda = 1 \pm 2i \]
These complex eigenvalues indicate an oscillatory solution, where:

• Solutions involve exponential functions multiplied by sinusoidal functions indicating oscillations with a certain frequency and growth/decay rate.
• This type of solution explains phenomena such as electrical circuits or mechanical vibrations.

Understanding complex eigenvalues provides insights into such systems' dynamic or periodic nature.

The characteristic equation is derived from setting the determinant of \(A - \lambda I \) to zero. This equation helps in finding eigenvalues of matrix \(A\). Through our worked example:

• Matrix subtraction results in \[\begin{pmatrix} -5 - \lambda & 10 \ -4 & 7 - \lambda \end{pmatrix} \]
• Next, calculate the determinant, simplifying to a quadratic equation, \[\lambda^2 - 2\lambda + 5 = 0 \]
• Solve using the quadratic formula: \[\lambda = \frac{2 \pm 4i}{2} \ = 1 \pm 2i \]
  This process pinpoints the eigenvalues critical to analyzing and solving the system.

  In summary, solving for the characteristic equation lets you decipher every eigenvalue influencing the system's solution behavior.