Question

Consider the given initial value problem \(\mathbf{y}^{\prime}=A \mathbf{y}, \mathbf{y}\left(t_{0}\right)=\mathbf{y}_{0}\). (a) Find the eigenvalues and eigenvectors of the coefficient matrix \(A\). (b) Construct a fundamental set of solutions. (c) Solve the initial value problem.\(\mathbf{y}^{\prime}=\left[\begin{array}{rr}-4 & 1 \\ -1 & -2\end{array}\right] \mathbf{y}, \quad \mathbf{y}(-1)=\left[\begin{array}{l}1 \\\ 0\end{array}\right]\)

Short answer

Question: Solve the given initial value problem (IVP) for a 2x2 linear system of first-order differential equations with matrix A and initial condition: Matrix A: \(\left[\begin{array}{rr}-4 & 1 \\\ -1 & -2\end{array}\right]\), Initial condition: \(\mathbf{y}(-1)=\begin{bmatrix}1\\0\end{bmatrix}\). Answer: The solution of the given IVP is: \(\mathbf{y}(t) = \frac{2}{e - e^{-5}} e^{-1t} \begin{bmatrix}1\\1\end{bmatrix} + \frac{e-1}{e - e^{-5}} e^{-5t} \begin{bmatrix}1\\-1\end{bmatrix}\).

Step-by-step solution

Find the eigenvalues of matrix A

First, we find the eigenvalues of the given coefficient matrix A: \(\left[\begin{array}{rr}-4 & 1 \\\ -1 & -2\end{array}\right]\). Calculate the determinant of the matrix \((A-\lambda I)\), then solve for \(\lambda\): \(\det(A-\lambda I) = (-4-\lambda)((-2)-\lambda) - (-1)(1) = \lambda^2 + 6\lambda + 7\implies \lambda_1 = -1,\, \lambda_2 = -5\).

Find the eigenvectors for each eigenvalue

For each eigenvalue, plug it into \((A-\lambda I) \mathbf{v} = \mathbf{0}\) to find the eigenvectors. For \(\lambda_1 = -1\), \((A-\lambda_1 I) = \left[\begin{array}{rr}-3&1 \\\ -1&-1\end{array}\right]\). Solving the system yields: \(\mathbf{v_1} = \begin{bmatrix}1\\1\end{bmatrix}\). For \(\lambda_2 = -5\), \((A-\lambda_2 I) = \left[\begin{array}{rr}1&1 \\\ -1&3\end{array}\right]\). Solving the system yields: \(\mathbf{v_2} = \begin{bmatrix}1\\-1\end{bmatrix}\).

Construct the fundamental set of solutions

We now have the eigenvalues and eigenvectors, so we can construct the fundamental set of solutions for the given initial value problem. The general solution is given by: \(\mathbf{y}(t) = c_1 e^{\lambda_1 t} \mathbf{v_1} + c_2 e^{\lambda_2 t} \mathbf{v_2}\). Substituting the eigenvalues and eigenvectors, we get: \(\mathbf{y}(t) = c_1 e^{-1t} \begin{bmatrix}1\\1\end{bmatrix} + c_2 e^{-5t} \begin{bmatrix}1\\-1\end{bmatrix}\).

Solve the initial value problem

Now we apply the initial condition \(\mathbf{y}(-1)=\begin{bmatrix}1\\0\end{bmatrix}\) to find the constants \(c_1\) and \(c_2\). Plugging in \(t=-1\) and the initial condition, we get: \(\begin{bmatrix}1\\0\end{bmatrix} = c_1 e \begin{bmatrix}1\\1\end{bmatrix} + c_2 e^{5} \begin{bmatrix}1\\-1\end{bmatrix}\). Solving this system of linear equations for \(c_1\) and \(c_2\), we obtain: \(c_1 = \frac{2}{e - e^{-5}}\), \(c_2 = \frac{e - 1}{e - e^{-5}}\). Finally, we write down the solution for the initial value problem: \(\mathbf{y}(t) = \frac{2}{e - e^{-5}} e^{-1t} \begin{bmatrix}1\\1\end{bmatrix} + \frac{e-1}{e - e^{-5}} e^{-5t} \begin{bmatrix}1\\-1\end{bmatrix}\).

Key concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, which play a pivotal role in solving systems of differential equations. An eigenvalue of a square matrix A is a scalar λ for which there is a non-zero vector v, known as an eigenvector, such that Av = λv. In other words, multiplying the matrix by the eigenvector only scales the eigenvector by the eigenvalue, and does not change its direction.

To find eigenvalues, we solve the characteristic equation det(A - λI) = 0, where I is the identity matrix. Once the eigenvalues are known, the corresponding eigenvectors can be found by solving (A - λI)v = 0 for each eigenvalue. It's worth noting that finding eigenvectors often involves solving a system of linear equations, which, in this case, can lead to a set of basic vectors from which all other vectors in the space can be derived.

Fundamental Set of Solutions

In the context of solving linear differential equations, a fundamental set of solutions refers to a set of independent solutions which can be combined to form the general solution of a differential equation. If we're dealing with an nth-order differential equation, then we need n independent solutions to make up a fundamental set.

Once we have the eigenvalues and eigenvectors from the coefficient matrix, we can use them to construct the fundamental set of solutions. Each eigenvalue-eigenvector pair contributes an independent solution of the form c_i e^λ_it v_i, where c_i are arbitrary constants. The superposition of these solutions provides the means to satisfy various initial conditions.

Coefficient Matrix

The coefficient matrix in a system of linear differential equations prominently dictates how the system behaves over time. For the initial value problem involving differential equations, the coefficient matrix A provides the connection between derivatives of the vector function y and the function itself, in the form y' = Ay.

The elements of the coefficient matrix are crucial—they influence the eigenvalues and the behavior of the system's solutions. For instance, stable systems tend to have matrices with eigenvalues having negative real parts. Observing the coefficient matrix's pattern can also indicate the potential symmetries or conservation laws inherent in the system being studied.

Differential Equations

Differential equations are equations that relate a function to its derivatives. They emerge ubiquitously in the sciences for modeling dynamic systems like populations, heat flow, or motion. Solving a differential equation means finding a function (or set of functions) that satisfies the relationship prescribed by the equation over a certain domain.

In our example, we're dealing with a system of linear first-order differential equations, which is solved using a matrix approach. The solution to this system requires integrating exponential functions, as the derivative of an exponential function is proportional to the function itself. This property makes exponential functions naturally suited for describing the evolution of systems governed by first-order linear differential equations, and it's the reason why solutions often involve e^λt terms, where λ is an eigenvalue of the system.