Problem 21 - Differential Equations and Linear Algebra

Question

Use the ideas introduced in this section to solve the given system of differential equations. $$x_{1}^{\prime}=-12 x_{1}-7 x_{2}, \quad x_{2}^{\prime}=16 x_{1}+10 x_{2}$$

Step-by-Step Solution

Answer

The solution of the given system of differential equations is: \(x_1(t) = C_1 e^{2t} + C_2 e^{8t}\) \(x_2(t) = 2C_1 e^{2t} - 4C_2 e^{8t}\)

Step 1: Convert to matrix form

The given system of differential equations can be expressed as: \(\begin{bmatrix} x_1' \\ x_2' \end{bmatrix} = \begin{bmatrix} -12 & -7 \\ 16 & 10 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\) Let the matrix A be: \[ A = \begin{bmatrix} -12 & -7 \\ 16 & 10 \end{bmatrix} \] Now we have: \(\begin{bmatrix} x_1' \\ x_2' \end{bmatrix} = A \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}\)

Step 2: Find the eigenvalues and eigenvectors of A

Find the eigenvalues of A by computing the determinant of \(A - \lambda I\), where \(\lambda\) is the eigenvalue, and I is the identity matrix. \(\det(A - \lambda I) = \begin{vmatrix} -12-\lambda & -7 \\ 16 & 10-\lambda \end{vmatrix} = (-12-\lambda)(10-\lambda) - (-7)(16)\) Expanding and solving for \(\lambda\): \((\lambda - 2)(\lambda - 8) = 0\) So, the eigenvalues are \(\lambda_1 = 2\) and \(\lambda_2 = 8\). Now find the corresponding eigenvectors for each eigenvalue by solving \((A-\lambda_i I) \mathbf{v} = 0\), where \(\mathbf{v}\) is the eigenvector. For \(\lambda_1 = 2\): \((A - 2I)\mathbf{v_1} = \begin{bmatrix} -14 & -7 \\ 16 & 8 \end{bmatrix}\mathbf{v_1} = 0\) Which gives \(\mathbf{v_1} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\) For \(\lambda_2 = 8\): \((A - 8I)\mathbf{v_2} = \begin{bmatrix} -20 & -7 \\ 16 & 2 \end{bmatrix}\mathbf{v_2} = 0\) Which gives \(\mathbf{v_2} = \begin{bmatrix} 1 \\ -4 \end{bmatrix}\)

Step 3: Write the general solution using eigenvalues and eigenvectors

Now, using the eigenvalues and eigenvectors found above, the general solution for the system of differential equations can be written as: \(\begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = C_1 e^{\lambda_1 t}\mathbf{v_1} + C_2 e^{\lambda_2 t}\mathbf{v_2}\) Substitute the values of eigenvalues and eigenvectors in the expression: \(\begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = C_1 e^{2t}\begin{bmatrix} 1 \\ 2 \end{bmatrix} + C_2 e^{8t}\begin{bmatrix} 1 \\ -4 \end{bmatrix}\) Therefore, the solution of the given system of differential equations is: \(x_1(t) = C_1 e^{2t} + C_2 e^{8t}\) \(x_2(t) = 2C_1 e^{2t} - 4C_2 e^{8t}\)

Key Concepts

system of differential equations

In mathematics, a **system of differential equations** involves two or more equations. These equations relate multiple functions with their rates of change. In this problem, we are given:

• \(x_{1}^{\prime} = -12x_1 - 7x_2\)
• \(x_{2}^{\prime} = 16x_1 + 10x_2\)

This system signifies how each variable \(x_1\) and \(x_2\) changes over time (represented by their derivatives \(x_{1}^{\prime}\) and \(x_{2}^{\prime}\)). Solving such a system gives us expressions for \(x_1(t)\) and \(x_2(t)\) over time, helping us understand the dynamics influenced by the coefficients (like \(-12, -7, 16, and 10\)) present in the equations.
Systems of differential equations frequently model real-world phenomena where multiple interacting elements evolve together, such as in predator-prey ecosystems, electrical circuits, or economic systems.

eigenvalues and eigenvectors

The concepts of **eigenvalues and eigenvectors** stem from the solution of systems of linear equations and play a crucial role in solving systems of differential equations. To find eigenvalues, we solve for \(\lambda\) in the equation:\[\text{det}(A - \lambda I) = 0\]Where \(A\) is our coefficient matrix, and \(I\) is the identity matrix of the same size as \(A\).
For this problem, we find that:

• Eigenvalues are \(\lambda_1=2\) and \(\lambda_2=8\).

Once eigenvalues are calculated, eigenvectors are found by solving:\[(A-\lambda_i I) \mathbf{v} = 0\]For each eigenvalue \(\lambda_i\). These vectors \(\mathbf{v_1}\) and \(\mathbf{v_2}\) correspond to \(\lambda_1\) and \(\lambda_2\), providing directions for growth or decay in solutions.

matrix form

The **matrix form** is a powerful way to express systems of differential equations. By representing the equations as matrices, we simplify the problem. The given system is:\[\begin{bmatrix} x_1' \ x_2' \end{bmatrix} = \begin{bmatrix} -12 & -7 \ 16 & 10 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix}\]Here, the left matrix represents the derivatives \([x_1', x_2']^T\), and the right matrix encapsulates the coefficients of the system.
This compact form allows us to use linear algebra techniques, such as finding eigenvalues and eigenvectors, which are instrumental in determining the behavior of the system over time. Matrix notation not only simplifies calculations but also provides insight into the inherent properties and solutions of the system.

general solution of differential equations

The **general solution of differential equations** gives a comprehensive expression for how variables change over time within a system. With known eigenvalues and eigenvectors, the solution for our system is expressed as:\[\begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix} = C_1 e^{\lambda_1 t}\mathbf{v_1} + C_2 e^{\lambda_2 t}\mathbf{v_2}\]For this problem, substituting \(\lambda_1 = 2\), \(\lambda_2 = 8\), \(\mathbf{v_1} = \begin{bmatrix} 1 \ 2 \end{bmatrix}\), and \(\mathbf{v_2} = \begin{bmatrix} 1 \ -4 \end{bmatrix}\), we get:

• \(x_1(t) = C_1 e^{2t} + C_2 e^{8t}\)
• \(x_2(t) = 2C_1 e^{2t} - 4C_2 e^{8t}\)

This solution depends on constants \(C_1\) and \(C_2\), which are determined by initial conditions. The expressions showcase how each part of the solution grows or decays over time, allowing us to predict system behavior and aid in practical applications.