Question

Find the general solution of each of the linear systems in Exercises 1-26. $$ \begin{aligned} &x^{\prime}=3 x-2 y \\ &y^{\prime}=2 x+3 y \end{aligned} $$

Short answer

The general solution of the given system of linear first-order differential equations is: \( \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = c_1 e^{(3 + 2i) t} (1, 1-i) + c_2 e^{(3 - 2i) t} (1, 1+i) \)

Step-by-step solution

Convert the system to matrix form

To convert the given system of linear first-order differential equations into a matrix form, we can write it as: \( \begin{pmatrix} x'\\ y' \end{pmatrix} = \begin{pmatrix} 3 & -2 \\ 2 & 3 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} \) Let's call the given matrix \(A\): \(A = \begin{pmatrix} 3 & -2 \\ 2 & 3 \end{pmatrix} \)

Calculate the eigenvalues and eigenvectors

To find the eigenvalues, calculate the characteristic polynomial of the matrix \(A\) as follows: \( det(A-\lambda I) = \begin{vmatrix} 3-\lambda & -2 \\ 2 & 3-\lambda \end{vmatrix} = (3-\lambda)^2 - (-2)(2) = 0 \) We get the characteristic polynomial as \(\lambda^2-6\lambda+13=0\). Now, we can find the eigenvalues \(\lambda_1\) and \(\lambda_2\). The eigenvalues are \(\lambda_1 = 3 + 2i\) and \(\lambda_2 = 3 - 2i\). Now, let's find the corresponding eigenvectors. For eigenvalue \(\lambda_1 = 3 + 2i\), we need to solve the following system: \( \begin{pmatrix} (3 - (3 + 2i)) & -2 \\ 2 & (3 - (3 + 2i)) \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \end{pmatrix} \) which simplifies to: \( \begin{pmatrix} -2i & -2 \\ 2 & -2i \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \end{pmatrix} \) We find the eigenvector corresponding to \(\lambda_1\) is \(v_1 = (1, 1 - i)\). Similarly, for eigenvalue \(\lambda_2 = 3 - 2i\), we get the eigenvector \(v_2 = (1, 1 + i)\).

Construct the general solution

Using the eigenvalues \(\lambda_1\) and \(\lambda_2\) and their corresponding eigenvectors, we construct the general solution in the form: \( \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = c_1 e^{\lambda_1 t} v_1 + c_2 e^{\lambda_2 t} v_2 \) where \(c_1\) and \(c_2\) are arbitrary constants. Substitute the eigenvalues and eigenvectors into the general solution: \( \begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = c_1 e^{(3 + 2i) t} (1, 1-i) + c_2 e^{(3 - 2i) t} (1, 1+i) \) This is the general solution of the given system of linear first-order differential equations.

Key concepts

Eigenvalues and Eigenvectors

To solve systems of linear differential equations using matrix methods, understanding eigenvalues and eigenvectors is crucial. These concepts help us analyze and simplify complex systems.

**Eigenvalues** are special numbers associated with a matrix. For a given square matrix \( A \), the eigenvalues \( \lambda \) satisfy the equation \( det(A - \lambda I) = 0 \), where \( I \) is the identity matrix. In our example, solving \( det(A - \lambda I) = 0 \) gives us the eigenvalues \( \lambda_1 = 3 + 2i \) and \( \lambda_2 = 3 - 2i \).

**Eigenvectors** are vectors that, when transformed by the matrix \( A \), do not change direction. To find them, we solve \( (A - \lambda I) \mathbf{v} = \mathbf{0} \) for each eigenvalue. These eigenvectors, such as \( \mathbf{v}_1 = (1, 1-i) \) and \( \mathbf{v}_2 = (1, 1+i) \) in our solution, help form the building blocks for the general solution of differential equations.

Matrix Form

Linear systems of differential equations can be intimidating, but matrix form simplifies this complexity. By representing the system in matrix form, we can easily manipulate and solve these equations.

In our example, the given system of equations:

• \(x' = 3x - 2y\)
• \(y' = 2x + 3y\)

is expressed as:
\[ \begin{pmatrix} x'\ y' \end{pmatrix} = \begin{pmatrix} 3 & -2 \ 2 & 3 \end{pmatrix} \begin{pmatrix} x\ y \end{pmatrix} \] Here, the matrix \( A = \begin{pmatrix} 3 & -2 \ 2 & 3 \end{pmatrix} \) captures the relationships between the variables and their derivatives. This matrix allows us to explore the dynamics of the system effectively.

General Solution

Once we have the eigenvalues and eigenvectors, constructing the general solution of a linear differential system becomes straightforward. This solution describes the behavior of the system over time and is pivotal in understanding the underlying dynamics.

The general solution is written as:
\[ \begin{pmatrix} x(t) \ y(t) \end{pmatrix} = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2 \] where \( c_1 \) and \( c_2 \) are constants determined by initial conditions. Substituting our previously calculated eigenvalues and eigenvectors, we get:
\[ \begin{pmatrix} x(t) \ y(t) \end{pmatrix} = c_1 e^{(3 + 2i)t} \begin{pmatrix} 1 \ 1-i \end{pmatrix} + c_2 e^{(3 - 2i)t} \begin{pmatrix} 1 \ 1+i \end{pmatrix} \] This solution helps us understand how the system evolves and responds to various conditions.