Question

Solve the initial value problem. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rrr} 4 & -8 & -4 \\ -3 & -1 & -3 \\ 1 & -1 & 9 \end{array}\right] \mathbf{y}, \quad \mathbf{y}(0)=\left[\begin{array}{r} -4 \\ 1 \\ -3 \end{array}\right] $$

Short answer

Question: Find the solution of the initial value problem with a linear system of ODEs in matrix form and an initial condition. Answer: To find the solution, follow these steps: 1. Find the eigenvalues and eigenvectors of the given matrix. 2. Write down the general solution of the linear system of ODEs. 3. Use the initial condition to find the unknown constants in the general solution. 4. Combine everything into the final solution of the initial value problem.

Step-by-step solution

Find the eigenvalues and eigenvectors of the matrix

Calculate the eigenvalues (which are the roots of the characteristic equation) of the matrix by solving the equation \(\text{det}(A - \lambda I) = 0\), where \(A\) is the given matrix and \(I\) is the identity matrix. Once you find the eigenvalues, \(\lambda_1, \lambda_2, \lambda_3\), find their respective eigenvectors by solving the equation \((A - \lambda_iI)\textbf{v}_i = 0\).

Write down the general solution of the linear system of ODEs

Use the eigenvalues and their corresponding eigenvectors to write the general solution of the linear system of ODEs: \(\mathbf{y}(t) = c_1e^{\lambda_1t}\mathbf{v}_1 + c_2e^{\lambda_2t}\mathbf{v}_2 + c_3e^{\lambda_3t}\mathbf{v}_3\), where \(c_1, c_2, c_3\) are constants to be determined.

Use the initial condition to find the unknown constants in the general solution

Plug the initial condition \(\mathbf{y}(0) = \left[\begin{array}{r} -4 \\ 1 \\ -3 \end{array}\right]\) into the general solution to determine the constants \(c_1, c_2, c_3\). Since \(\mathbf{y}(0) = c_1e^{\lambda_1 \cdot 0}\mathbf{v}_1 + c_2e^{\lambda_2 \cdot 0}\mathbf{v}_2 + c_3e^{\lambda_3 \cdot 0}\mathbf{v}_3 = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + c_3\mathbf{v}_3\), we can solve the system of equations to find \(c_1, c_2, c_3\).

Combine everything into the final solution of the initial value problem

After finding the constants \(c_1, c_2, c_3\), use them to form the complete solution of the initial value problem by substituting them back into the general solution: \(\mathbf{y}(t) = c_1e^{\lambda_1t}\mathbf{v}_1 + c_2e^{\lambda_2t}\mathbf{v}_2 + c_3e^{\lambda_3t}\mathbf{v}_3\).

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is crucial in solving linear systems of ordinary differential equations (ODEs). These concepts help simplify complex systems and find solutions efficiently.
To find eigenvalues, we need to solve the characteristic equation:

• We subtract \(\lambda\) times the identity matrix, \(I\), from the given matrix, \(A\).
• Then, calculate the determinant of this new matrix, \(\text{det}(A - \lambda I)\).
• The solutions to \(\text{det}(A - \lambda I) = 0\) give us the eigenvalues.

Once you have the eigenvalues, \(\lambda_1, \lambda_2, \lambda_3\), the next step is to find their corresponding eigenvectors. This involves solving \((A - \lambda_iI)\mathbf{v}_i = 0\) for each \(\lambda_i\). Eigenvectors show the direction of the vector transformation defined by \(A\).

Linear System of ODEs

A linear system of ordinary differential equations can model complex systems involving multiple interrelated components, like in physical, biological, or engineering contexts.
In our case, the system \(\mathbf{y}^{\prime} = A\mathbf{y}\) represents such a model, where \(A\) is a 3x3 matrix affecting the function's behavior over time.
By expressing the system in matrix form, it allows us to use tools like eigenvalues and eigenvectors for simplifying and solving the equations.
This system of ODEs leads us to the eigenvectors and eigenvalues that help form solutions of the system. Finding these values is the core initial step in tackling these types of problems effectively.

Characteristic Equation

The characteristic equation plays a fundamental role in obtaining eigenvalues for the linear system's matrix.
The process involves several steps:

• Form the matrix \(A - \lambda I\), where \(\lambda\) is a scalar that we apply to each element of the identity matrix \(I\).
• Calculate the determinant of this matrix, \(\text{det}(A - \lambda I)\), to form a polynomial equation.
• The roots of this polynomial provide the eigenvalues, which are vital in constructing the system's solutions.

Eigenvalues derived from the characteristic equation help us know how the system behaves in terms of stability and oscillation patterns.

General Solution of Differential Equations

Once eigenvalues and eigenvectors are determined, they contribute to the general solution of the differential equation system.
The general solution \(\mathbf{y}(t)\) is expressed as a combination of exponential functions based on the eigenvalues:

• \(\mathbf{y}(t) = c_1e^{\lambda_1t}\mathbf{v}_1 + c_2e^{\lambda_2t}\mathbf{v}_2 + c_3e^{\lambda_3t}\mathbf{v}_3\) involves multiplying each eigenvector by an exponential function of its corresponding eigenvalue \(\lambda_i\).
• The constants \(c_1, c_2, c_3\) are determined using initial conditions that confirm the solution fits the specifics of the problem.

This technique generalizes to any size of systems and helps resolve complex interdependent systems in linear algebra and differential equations.