Solve the ODE(x1(t) x2(t))′=(2 −5 1 −2) (x1(t) x2(t)) 

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.

Solve the ODE (x1(t) x2(t))′=(2 −5 1 −2) (x1(t) x2(t))

Learn how to solve the system of ODEs (x1(t) x2(t))′=(2 −5 1 −2) (x1(t) x2(t)) by finding eigenvalues and eigenvectors of the matrix. This solution explores complex eigenvalues, eigenvectors, and the resulting oscillatory solutions, covering key linear algebra and differential equation concepts for undergraduates.

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