Question

Find the general solution. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} -7 & 4 \\ -1 & -11 \end{array}\right] \mathbf{y} $$

Short answer

Eigenvalues: λ1 = -9 + 2i and λ2 = -9 - 2i Eigenvectors: For λ1, v1 = k1 * [2i, 1 + 2i]^T For λ2, v2 = k2 * [-2i, 1 - 2i]^T

Step-by-step solution

Find the eigenvalues of the matrix

To find the eigenvalues of the given matrix, we need to solve the following equation: $$\det(A - \lambda I) = 0$$ Where A is the given matrix, λ is the eigenvalue, and I is the identity matrix. In our case: $$\det\left(\begin{bmatrix}-7-\lambda & 4 \\ -1 & -11-\lambda \end{bmatrix}\right)= 0$$ Now, let's compute the determinant and solve the equation: $$( -7 - \lambda ) ( -11 - \lambda ) - ( 4 ) ( -1 ) = 0$$ $$\lambda^2 + 18 \lambda + 77 = 0$$ This equation has complex solutions λ1 = -9 + 2i and λ2 = -9 - 2i.

Find the eigenvectors corresponding to each eigenvalue

For each eigenvalue, we need to solve the following equation to find the eigenvector: $$ (A - \lambda I) \mathbf{v} = 0 $$ For λ1 = -9 + 2i : $$(A - (-9 + 2i) I) \mathbf{v}_1 = \begin{bmatrix}2-2i & 4 \\ -1 & -2-2i \end{bmatrix}\mathbf{v}_1 = 0$$ Solving the system, we get the eigenvector: $$\mathbf{v}_1 = k_1 \begin{bmatrix}2i \\ 1 + 2i \end{bmatrix}$$ For λ2 = -9 - 2i : $$(A - (-9 - 2i) I) \mathbf{v}_2 = \begin{bmatrix}2+2i & 4 \\ -1 & -2+2i \end{bmatrix}\mathbf{v}_2 = 0$$ Solving the system, we get the eigenvector: $$\mathbf{v}_2 = k_2 \begin{bmatrix}-2i \\ 1 - 2i \end{bmatrix}$$

Form the general solution

Now, using the eigenvalues and eigenvectors, we can form the general solution for the given system of differential equations: $$\mathbf{y}(t) = c_1 e^{(-9+2i)t} \begin{bmatrix}2i \\ 1 + 2i \end{bmatrix} + c_2 e^{(-9-2i)t} \begin{bmatrix}-2i \\ 1 - 2i \end{bmatrix}$$ This is the general solution to the given system of differential equations.

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is essential for solving many problems in mathematics and physics, especially when dealing with linear transformations and systems of differential equations.
Eigenvalues are special numbers associated with a square matrix. We find them by solving the characteristic polynomial, which is obtained from the determinant of the matrix subtracted by the eigenvalue multiplied by the identity matrix. The solutions to this equation are the eigenvalues and they give us important information about the matrix, such as its invertibility or stability properties.
Once we have the eigenvalues, we search for eigenvectors, which are non-zero vectors that, when multiplied by the matrix, result in a vector parallel to the original one, just scaled by the corresponding eigenvalue. Essentially, the eigenvector gets stretched or shrunk by the matrix, but its direction remains unchanged.

• Eigenvalues describe how much the direction of eigenvector is stretched.
• Eigenvectors maintain their direction but are scaled by their respective eigenvalues when transformed by the matrix.

In our exercise, complex eigenvalues appear, indicating oscillatory solutions for the system, which is typical for systems involving rotations or periodic motion, reflecting the real-world dynamics such as electrical circuits or mechanical systems.

System of Differential Equations

A system of differential equations consists of multiple equations involving functions and their derivatives. This type of system can model a wide range of phenomena, from population dynamics to mechanical systems.
In the provided exercise, we have a linear system of differential equations with constant coefficients represented in matrix form. To solve it, we look for eigenvalues and eigenvectors of the coefficient matrix. These provide us the structure of the solution to the system.

• We solve for eigenvalues to understand the nature of the solution, whether it is oscillatory, exponential growth or decay.
• The corresponding eigenvectors offer us the direction and the form of the solution.
• Complex eigenvalues typically indicate oscillatory behavior, translated into trigonometric functions in the solution.

Learning to solve such systems is crucial for students as it empowers them to handle real-life problems where several interconnected variables evolve over time, such as in engineering or economics.

Complex Numbers

Complex numbers, comprising a real and an imaginary part, are a fundamental concept in mathematics, especially when dealing with systems that involve oscillations or waves.
In the context of differential equations, complex numbers often emerge as eigenvalues of a system, especially when the system has oscillatory solutions. For instance, in electrical engineering, the behavior of circuits with inductors and capacitors can lead to complex eigenvalues, indicating a sinusoidal response.

• Complex numbers are written in the form a + bi, where a is the real part, b is the coefficient of the imaginary part, and i is the square root of -1.
• The real part of a complex eigenvalue typically reflects damping or growth in the system, while the imaginary part represents oscillation.

Working with complex numbers can seem daunting at first, but they are just an extension of the real number system, enabling us to solve problems that cannot be addressed using only real numbers, like the differential equations in our exercise.