To solve the given system of first-order linear ordinary differential equations, let’s write it out in vector form. The system is given by:

$\frac{d}{dt} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix} \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix}.$

This is a system of linear differential equations, which can be solved by finding the eigenvalues and eigenvectors of the matrix $A = \begin{pmatrix} 2 & -5 \\ 1 & -2 \end{pmatrix}$.

Step 1: Find the Eigenvalues of $A$

The eigenvalues $\lambda$ of matrix $A$ are found by solving the characteristic equation:

$\det(A - \lambda I) = 0.$

1. Calculate $A - \lambda I$:

   $A - \lambda I = \begin{pmatrix} 2 - \lambda & -5 \\ 1 & -2 - \lambda \end{pmatrix}.$

2. Calculate the determinant:

   $\det(A - \lambda I) = (2 - \lambda)(-2 - \lambda) - (-5)(1).$

   Expanding this, we get:

   $\lambda^2 - 2\lambda + 5 = 0.$

This characteristic polynomial has complex roots:

$\lambda = -1 \pm i\sqrt{5}.$

Step 2: Find the Eigenvectors

For each eigenvalue $\lambda = -1 + i\sqrt{5}$ and $\lambda = -1 - i\sqrt{5}$, we can find the corresponding eigenvector.

Step 3: General Solution

With complex eigenvalues, the solution will be of the form:

$\mathbf{x}(t) = e^{\alpha t} \left( C_1 \cos(\beta t) \mathbf{v}_1 + C_2 \sin(\beta t) \mathbf{v}_2 \right),$

where $\alpha = -1$, $\beta = \sqrt{5}$, and $\mathbf{v}_1$ and $\mathbf{v}_2$ are the real and imaginary parts of the eigenvector associated with $\lambda = -1 + i\sqrt{5}$.

Final Solution

The general solution is given by:

$\begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} = e^{-t} \left( C_1 \cos(\sqrt{5} t) \begin{pmatrix} 1 \\ \frac{1}{2} \end{pmatrix} + C_2 \sin(\sqrt{5} t) \begin{pmatrix} 1 \\ \frac{1}{2} \end{pmatrix} \right).$

Let me know if you need further details or explanations!

Related Questions

1. How do we find eigenvalues and eigenvectors for a 2x2 matrix?
2. What does it mean for an ODE to have complex eigenvalues, and how does it affect the solution?
3. How do we interpret the terms $e^{\alpha t} \cos(\beta t)$ and $e^{\alpha t} \sin(\beta t)$ in the solution?
4. How can we determine stability of solutions for such a system of ODEs?
5. What is the physical interpretation of oscillatory solutions in dynamical systems?

Tip:

Always verify the stability of the system by examining the real part of the eigenvalues.