Question

Find the general solution. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rr} 3 & 1 \\ -1 & 1 \end{array}\right] \mathbf{y} $$

Short answer

Answer: The general form of the solution for a system of first-order linear differential equations is given by: $$\mathbf{y}(t) = c_1 \mathrm{e}^{\lambda_1 t} \mathbf{v_1} + c_2 \mathrm{e}^{\lambda_2 t} \mathbf{v_2}$$ where \(\lambda_1\) and \(\lambda_2\) are the eigenvalues, \(\mathbf{v_1}\) and \(\mathbf{v_2}\) are their corresponding eigenvectors, and \(c_1\) and \(c_2\) are arbitrary constants.

Step-by-step solution

Determine the eigenvalues

We start by finding the eigenvalues of the matrix. The eigenvalues, λ, can be obtained by solving the following equation, known as the characteristic equation: $$\det\left(\begin{array}{rr}3-\lambda & 1\\-1 & 1-\lambda\end{array}\right)=0$$ Expanding the determinant: $$ (3-\lambda)(1-\lambda) - (1 \cdot (-1)) = 0 $$ This results in a quadratic equation of λ: $$\lambda^2 - 4\lambda + 2 = 0$$ By solving this quadratic equation, we can find the eigenvalues λ.

Determine the eigenvectors

Once the eigenvalues are found, you must determine the eigenvectors (or generalized eigenvectors, if necessary) for each eigenvalue. The eigenvectors can be found by solving the system of homogeneous equations: $$(A-\lambda I)\mathbf{v} = 0$$ Where A is the given matrix, λ is an eigenvalue, and \(\mathbf{v}\) is the eigenvector we are trying to find. The eigenvectors of the given linear system are needed to construct the general solution.

Construct the general solution

Once you have the eigenvalues and the associated eigenvectors, you can construct the general solution by expressing the dependence of \(\mathbf{y}\) on the eigenvalues and eigenvectors. The general form of the solution will be: $$\mathbf{y}(t) = c_1 \mathrm{e}^{\lambda_1 t} \mathbf{v_1} + c_2 \mathrm{e}^{\lambda_2 t} \mathbf{v_2}$$ Where \(\lambda_1\) and \(\lambda_2\) are the eigenvalues obtained in step 1, \(\mathbf{v_1}\) and \(\mathbf{v_2}\) are their corresponding eigenvectors found in step 2, and \(c_1\) and \(c_2\) are arbitrary constants. Thus, by completing steps 1-3, you can find the general solution to the given system of linear differential equations.

Key concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are foundational concepts when studying linear transformations and systems of differential equations. In essence, an eigenvalue is a scalar that illustrates how much an eigenvector is stretched or squished during a transformation. An eigenvector is a nonzero vector that only changes by a scalar factor when a transformation is applied.
When solving differential equations, finding the eigenvalues helps determine the behavior of the system. For example, whether the system is stable or unstable and what the long-term behavior of solutions looks like. Solving for the eigenvalues requires calculating the determinant of the matrix minus a scalar multiple of the identity matrix. Once the eigenvalues are known, you can find the corresponding eigenvectors by solving a system of homogeneous equations. These eigenvectors are crucial because they provide the direction along which the solution to the differential equation evolves over time.
In the context of the given exercise, eigenvalues and eigenvectors allow us to build the general solution of the system of differential equations, which is a linear combination of scalar exponentials multiplied by the eigenvectors. This general solution aptly describes the reversible linear dynamics of the system.

Characteristic Equation

The characteristic equation is a fundamental part of understanding matrix transformations and their application in solving systems of linear differential equations. It arises from the need to find eigenvalues. We derive the characteristic equation from setting the determinant of the matrix after subtracting a scalar multiple of the identity matrix equal to zero. The resulting polynomial equation in terms of the scalar, typically denoted as \( \lambda \), contains the eigenvalues which are the roots of this equation.
To solve the characteristic equation, as portrayed in the example exercise, you will often end up with a polynomial whose degree corresponds to the number of eigenvalues of the matrix. By finding these eigenvalues, we gain insight into the system's behavior, allowing us to progress toward finding the system's general solution. Keep in mind that each eigenvalue corresponds to an eigenvector, and it’s these vectors that build the foundation for the solution to the differential equation.

Homogeneous Linear Systems

A homogeneous linear system is a system of linear equations where all of the constant terms are zero. These systems are significant in the study of linear algebra and differential equations. For the system to have nontrivial solutions (solutions other than the zero vector), the determinant of the coefficients matrix must be zero. This is directly related to finding the eigenvalues of a matrix.
Homogeneous systems are particularly interesting because their solutions can be expressed as combinations of eigenvectors, scaled by exponential functions of the eigenvalues and time variable in the context of differential equations. The key property of these systems is the principle of superposition, enabling us to add individual solutions together to obtain more complex solutions. In the example given, the homogeneity of the system is instrumental in linking the system's dynamics directly to the eigenvalues and the structure of its solutions. By finding the eigenvalues, eigenvectors, and using the superposition principle, one can construct a comprehensive general solution that encompasses all possible behaviors of the system over time.