Question

In Exercises \(17-24\) solve the initial value problem. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rrr} 4 & -4 & 4 \\ -10 & 3 & 15 \\ 2 & -3 & 1 \end{array}\right] \mathbf{y}, \quad \mathbf{y}(0)=\left[\begin{array}{r} 16 \\ 14 \\ 6 \end{array}\right] $$

Short answer

#Response# The specific solution for the given initial value problem can be found by following these steps: 1. Find the matrix exponential of the given matrix A, denoted by \(e^{At}\). 2. Find the eigenvalues and eigenvectors of matrix A. 3. Form a matrix E using the eigenvectors and a diagonal matrix D with corresponding eigenvalues on the diagonal. 4. Compute the matrix exponential \(e^{At}\) using the matrices E and D. 5. Determine the general solution of the system of linear differential equations using the matrix exponential: \(\mathbf{y}(t) = e^{At}\mathbf{c}\). 6. Apply the given initial condition \(\mathbf{y}(0)=\left[\begin{array}{r} 16 \\ 14 \\ 6 \end{array}\right]\) to find the specific solution for the problem.

Step-by-step solution

1. Find the matrix exponential of the given matrix

We're given the matrix $$ A = \left[\begin{array}{rrr} 4 & -4 & 4 \\ -10 & 3 & 15 \\ 2 & -3 & 1 \end{array}\right] $$ We need to find \(e^{At}\). In order to do that, we first find the eigenvalues and eigenvectors of \(A\).

2. Find the eigenvalues and eigenvectors of matrix A

Computing the eigenvalues of A, by finding the roots of the characteristic equation \(|A-\lambda I|=0\): $$ \begin{vmatrix} 4-\lambda & -4 & 4 \\ -10 & 3-\lambda & 15 \\ 2 & -3 & 1-\lambda \end{vmatrix} = 0 $$ Solving this equation, we find the eigenvalues \(\lambda_1=1\), \(\lambda_₂=2\), and \(\lambda_₃=5\). For each \(\lambda_i\), we substitute it into the equation \((A-\lambda_i I)v_i=0\) to find the eigenvectors \(v_i\).

3. Find the matrix E and diagonal matrix D

Form a matrix E by combining the eigenvectors \(v_1\), \(v_2\), and \(v_3\) (normalized, if necessary) in columns and create a diagonal matrix D with corresponding eigenvalues on the diagonal: $$ E=\left[\begin{array}{ccc} v_{1} & v_{2} & v_{3} \end{array}\right], \quad D=\left[\begin{array}{ccc} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & \lambda_{3} \end{array}\right] $$

4. Compute the matrix exponential of At

Using the matrix E and diagonal matrix D, compute the matrix exponential \(e^{At}=Ee^{Dt}E^{-1}\). Since D is a diagonal matrix, the matrix exponential \(e^{Dt}\) can be computed by taking the exponential of the diagonal elements: $$ e^{Dt} = \left[\begin{array}{ccc} e^{\lambda_{1}t} & 0 & 0 \\ 0 & e^{\lambda_{2}t} & 0 \\ 0 & 0 & e^{\lambda_{3}t} \end{array}\right] $$

5. Find the general solution

Now we have all the components needed to compute the general solution of the given system of linear differential equations, which is given by: $$ \mathbf{y}(t) = e^{At}\mathbf{c} = Ee^{Dt}E^{-1}\mathbf{c} $$ where \(\mathbf{c}\) is a column vector of constants.

6. Apply the initial condition

We are given the initial condition $\mathbf{y}(0)=\left[\begin{array}{r} 16 \\ 14 \\ 6 \end{array}\right]\(. To find the specific solution, substitute t=0 in the general solution, and solve for \)\mathbf{c}$ $$ \mathbf{y}(0) = Ee^{D(0)}E^{-1}\mathbf{c} $$ Solve for \(\mathbf{c}\) and substitute it into the general solution to obtain the specific solution for the given initial value problem.

Key concepts

Eigenvalues

In the context of solving a system of differential equations, finding eigenvalues is a crucial step. Eigenvalues are special numbers associated with a matrix that provide insight into the system's behavior. They are derived from the characteristic equation, a determinant equation like \(|A-\lambda I|=0\). Here, \A\ is your matrix and \lambda\ represents the eigenvalues. These eigenvalues \(\lambda_1, \lambda_2, \lambda_3\) tell us how the system evolves over time.

When worked out, they can help predict stability, oscillation, or growth patterns in the solutions of the differential equations. For example, a positive eigenvalue suggests growth, while a negative one implies decay. In our problem, we find eigenvalues \(\lambda_1=1, \lambda_2=2, \lambda_3=5\), which will play a direct role in solving the system.

Matrix Exponential

Matrix exponential can often seem tricky, but it is essential for solving linear differential systems. The matrix exponential \(e^{At}\) is a way to apply the exponential function to matrices. It is analogous to the normal exponential function but in the matrix context.

The purpose of the matrix exponential is to help find the solution to differential equations such as \(\mathbf{y}' = A\mathbf{y}\). Calculating \(e^{At}\) involves transforming the system matrix into a format where you can apply the exponential function—usually by using its eigenvalues and their corresponding eigenvectors, which simplify the calculation process.

With the diagonal matrix \(D\) containing the eigenvalues, \(e^{Dt}\) lets us easily compute the exponential by applying it only to the diagonal elements. This forms the backbone of finding the unique solution to the initial value problem.

Eigenvectors

Eigenvectors complement eigenvalues in understanding a matrix's properties and solving differential equations. For each eigenvalue, there is at least one corresponding eigenvector. These vectors \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) are non-zero vectors that satisfy the equation \( (A - \lambda_i I)\mathbf{v}_i = 0 \).

By solving for these vectors, you add direction and magnitude information to the corresponding eigenvalue, creating a fuller picture of your system's dynamics. The eigenvectors form the columns of the matrix \(E\), which is used to diagonalize the matrix and compute the matrix exponential.

Essentially, while eigenvalues tell us 'how fast,' eigenvectors tell us 'in which direction' the system's state changes and evolves over time.

System of Differential Equations

Differential equations often represent dynamic systems in physics, engineering, and other sciences. A system like \(\mathbf{y}' = A\mathbf{y}\) involves multiple interdependent equations and multiple variables captured in a vector format.

This system's goal is to find a vector function \(\mathbf{y}(t)\) that satisfies the equation. By doing so, you determine how the system's state evolves with time. \(A\) is a constant matrix representing the linear relationships between variables in the differential equation system.

Solving this type of system often involves calculating eigenvalues and eigenvectors, then using these to form a matrix exponential that describes the system's temporal dynamics under given initial conditions. This approach helps simplify complex systems into manageable components, making predictions and understanding easier.