Question

find a fundamental matrix for the given system of equations. In each case also find the fundamental matrix \(\mathbf{\Phi}(t)\) satisfying \(\Phi(0)=\mathbf{1}\) $$ \mathbf{x}^{\prime}=\left(\begin{array}{rr}{-\frac{3}{4}} & {\frac{1}{2}} \\\ {\frac{1}{8}} & {-\frac{1}{4}}\end{array}\right) \mathbf{x} $$

Short answer

Question: Find the fundamental matrix \(\mathbf{\Phi}(t)\) that satisfies \(\Phi(0) = \mathbf{1}\) for the following system of linear differential equations: $$ \begin{cases} x_1'(t) = -\frac{3}{4}x_1(t) + \frac{1}{2}x_2(t) \\ x_2'(t) = \frac{1}{8}x_1(t) - \frac{1}{4}x_2(t) \end{cases} $$ Answer: The fundamental matrix \(\mathbf{\Phi}(t)\) satisfying \(\Phi(0) = \mathbf{1}\) is: $$\mathbf{\Phi}(t) = \begin{pmatrix} 1-e^{-\frac{1}{2}t} & e^{-\frac{1}{4}t}(1-e^{-\frac{1}{4}t}) \\ 2e^{-\frac{1}{2}t}-e^{-\frac{1}{4}t} & 2e^{-\frac{1}{4}t}-e^{-\frac{1}{2}t} \end{pmatrix}$$

Step-by-step solution

Eigenvalue and Eigenvectors Calculation

First, in order to find the solutions to the given system of linear differential equations, we have to find the eigenvalues and eigenvectors for the matrix. Consider matrix A: $$ \mathbf{A} = \left(\begin{array}{rr}{-\frac{3}{4}} & {\frac{1}{2}} \\ {\frac{1}{8}} & {-\frac{1}{4}} \end{array}\right) $$ For finding the eigenvalues (\(\lambda\)), calculate the determinant of the matrix \(\mathbf{A} - \lambda \mathbf{I}\): $$ \det (\mathbf{A} - \lambda \mathbf{I}) = \det \left(\begin{array}{rr}{-\frac{3}{4}-\lambda} & {\frac{1}{2}} \\ {\frac{1}{8}} & {-\frac{1}{4}-\lambda} \end{array}\right) = \lambda^2 + \frac{1}{2}\lambda +\frac{1}{8} $$ Solve for \(\lambda\): $$ \lambda^2 + \frac{1}{2}\lambda + \frac{1}{8} = 0 $$ The eigenvalues are \(\lambda_1 = -\frac{1}{2}\) and \(\lambda_2 = -\frac{1}{4}\). Next, find the eigenvectors for each eigenvalue. For \(\lambda_1 = -\frac{1}{2}\): $$ \begin{cases} -\frac{3}{4}\xi_1 + \frac{1}{2}\xi_2 = \frac{1}{2}\xi_1 \\ \frac{1}{8}\xi_1 - \frac{1}{4}\xi_2 = \frac{1}{2}\xi_2 \end{cases} $$ We get the eigenvector \(\mathbf{\xi}_1 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}\) For \(\lambda_2 = -\frac{1}{4}\): $$ \begin{cases} -\frac{3}{4}\xi_1 + \frac{1}{2}\xi_2 = \frac{1}{4}\xi_1 \\ \frac{1}{8}\xi_1 - \frac{1}{4}\xi_2 = \frac{1}{4}\xi_2 \end{cases} $$ We get the eigenvector \(\mathbf{\xi}_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}\)

Find the Fundamental Matrix

Using the eigenvalues and eigenvectors, we can write down the fundamental matrix \(\mathbf{X}(t)\): $$\mathbf{X}(t) = \begin{pmatrix} 2e^{-\frac{1}{2}t} & e^{-\frac{1}{4}t} \\ e^{-\frac{1}{2}t} & -e^{-\frac{1}{4}t} \end{pmatrix}$$ Here, the columns are linearly independent solutions to the system of linear differential equations.

Finding the Fundamental Matrix \(\mathbf{\Phi}(t)\) with Initial Condition \(\Phi(0) = \mathbf{1}\)

To find the fundamental matrix \(\mathbf{\Phi}(t)\) that satisfies the initial condition \(\Phi(0) = \mathbf{1}\), evaluate the matrix \(\mathbf{X}(t)\) at \(t = 0\): $$\mathbf{X}(0) = \begin{pmatrix} 2 & 1 \\ 1 & -1 \end{pmatrix}$$ Now, find the inverse of \(\mathbf{X}(0)\): $$[\mathbf{X}(0)]^{-1} = \frac{1}{(\det \mathbf{X}(0))} \begin{pmatrix} -1 & -1 \\ -1 & 2 \end{pmatrix} = \ \begin{pmatrix} -1 & -1 \\ -1 & 2 \end{pmatrix}$$ The fundamental matrix \(\mathbf{\Phi}(t)\) can be found by multiplying the inverse of \(\mathbf{X}(0)\) with the matrix \(\mathbf{X}(t)\): $$\mathbf{\Phi}(t) = [\mathbf{X}(0)]^{-1} \mathbf{X}(t) = \begin{pmatrix} -1 & -1 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} 2e^{-\frac{1}{2}t} & e^{-\frac{1}{4}t} \\ e^{-\frac{1}{2}t} & -e^{-\frac{1}{4}t} \end{pmatrix} = \begin{pmatrix} 1-e^{-\frac{1}{2}t} & e^{-\frac{1}{4}t}(1-e^{-\frac{1}{4}t}) \\ 2e^{-\frac{1}{2}t}-e^{-\frac{1}{4}t} & 2e^{-\frac{1}{4}t}-e^{-\frac{1}{2}t} \end{pmatrix}$$ Thus, the fundamental matrix \(\mathbf{\Phi}(t)\) satisfying \(\Phi(0) = \mathbf{1}\) is: $$\mathbf{\Phi}(t) = \begin{pmatrix} 1-e^{-\frac{1}{2}t} & e^{-\frac{1}{4}t}(1-e^{-\frac{1}{4}t}) \\ 2e^{-\frac{1}{2}t}-e^{-\frac{1}{4}t} & 2e^{-\frac{1}{4}t}-e^{-\frac{1}{2}t} \end{pmatrix}$$

Key concepts

Eigenvalues and Eigenvectors

Understanding the role of eigenvalues and eigenvectors is crucial in solving systems of linear differential equations. In essence, eigenvalues provide insight into the properties of a transformation represented by a matrix, and their corresponding eigenvectors point towards the directions that remain unaltered by this transformation.

Eigenvalues (\( \text{\text{lambda}} \_i \) for the i-th eigenvalue) are found by solving the characteristic equation, which, in our exercise, is derived from the determinant of the difference between the matrix \( \text{\textbf{A}} \) and the identity matrix multiplied by \( \text{\text{lambda}} \) (referred to as \( \text{\textbf{A}} - \text{\text{lambda}} \_i \text{\textbf{I}} \)). The solutions to this equation are the eigenvalues of \( \text{\textbf{A}} \).

The eigenvectors are then calculated for each eigenvalue, indicating the 'directions' that stay invariant when the matrix \( \text{\textbf{A}} \) acts on them. These eigenvectors form the basis for the solution space of the differential equations. In our exercise, the eigenvectors were \( \text{\textbf{\text{\text{xi}}}} \_1 \) for \( \text{\text{lambda}} \_1 \) and \( \text{\textbf{\text{\text{xi}}}} \_2 \) for \( \text{\text{lambda}} \_2 \), which created the fundamental matrix \( \text{\textbf{X}}(t) \).

It's important for students to understand that these eigenvalues and eigenvectors are not just abstract concepts—they're the backbone of solving practical problems involving systems of differential equations.

Linear Differential Equations

Linear differential equations form the bedrock of many mathematical models in physics, engineering, and various scientific fields. These equations describe the relation between a function and its derivatives.

In the realm of systems of linear differential equations, like the one in our exercise, solutions can be expressed using linear combinations of functions. A fundamental matrix \( \text{\textbf{X}}(t) \) is a matrix whose columns are linearly independent solutions to the differential system. The fundamental matrix is pivotal because it helps us to not just have singular solutions but encapsulates the entire set of solutions to the system.

Importance of finding a fundamental matrix

In our example, \( \text{\textbf{X}}(t) \) reflects two individual solutions to the system being combined into one matrix representation. The significance of \( \text{\textbf{X}}(t) \) lies in its ability to link the initial state of a system to its state at any later time t, essentially describing the evolution of the system over time.

The solutions are typically exponentials incorporated with eigenvalues, indicative of the systems' behavior, whether it's dampening oscillations or growth-decay patterns. Grasping the formulation and implications of the fundamental matrix is essential for comprehending how myriad systems behave dynamically.

Initial Condition Problem

Solving an initial condition problem is like being given a starting point and then tracing out the path that a system will follow. For differential equations, initial conditions are necessary to identify a specific solution among the infinite pool of possibilities that satisfy the equations.

The fundamental matrix \( \text{\textbf{\text{\text{Phi}}}}(t) \) we find must coincide with these initial conditions to be considered a valid solution. In our exercise, the initial condition was that \( \text{\textbf{\text{\text{Phi}}}}(0) \) equals the identity matrix \( \text{\textbf{1}} \)—a common requirement that ensures the solution matrix aligns with the state of the system at time zero.

Transforming to meet initial conditions

The transformation that the fundamental matrix needs to undergo involves finding the inverse of the matrix evaluated at time zero and multiplying it to the general solution. This step ensures that when we evaluate \( \text{\textbf{\text{\text{Phi}}}}(t) \) at time t=0, we retrieve the identity matrix, satisfying the initial conditions. In our solution steps, we computed \( [\text{\textbf{X}}(0)]^{-1} \) and used it to transform \( \text{\textbf{X}}(t) \) into \( \text{\textbf{\text{\text{Phi}}}}(t) \) that adheres to the prescribed starting point.

Understanding initial condition problems not only helps in finding specific solutions but also in appreciating the uniqueness a single set of initial conditions can bring to a system's behavior over time.