Question

In Exercises \(78-88\) find a fundamental set of solutions to \(\mathbf{x}^{\prime}=A \mathbf{x}\). Solve the initial value problem with \(\mathbf{x}(0)=\mathbf{x}_{0}\). $$ A=\left[\begin{array}{rrrr} 17 & -9 & 14 & 51 \\ 12 & -6 & 9 & 34 \\ 16 & -11 & 14 & 50 \\ -8 & 5 & -7 & -25 \end{array}\right], \quad \mathbf{x}_{0}=\left[\begin{array}{r} -2 \\ -2 \\ 1 \\ 0 \end{array}\right] $$

Short answer

Find eigenvalues and eigenvectors for matrix A; determine the constants to satisfy the initial condition using the fundamental solutions.

Step-by-step solution

Find Eigenvalues of A

To find a fundamental set of solutions, we need to determine the eigenvalues of matrix \(A\). We do this by solving the characteristic equation \(\det(\lambda I - A) = 0\), where \(I\) is the identity matrix of the same size as \(A\). Compute the determinant \(\det(\lambda I - A) = 0\) and solve for \(\lambda\).

Find Eigenvectors for Each Eigenvalue

Once the eigenvalues \(\lambda_1, \lambda_2, ..., \lambda_n\) are found, for each eigenvalue \(\lambda_i\), solve the equation \((A - \lambda_i I)\mathbf{v} = 0\) to find the corresponding eigenvector \(\mathbf{v}_i\). These will form the set of eigenvectors that correspond to the eigenvalues found in the previous step.

Form Fundamental Set of Solutions

The fundamental set of solutions to the system \(\mathbf{x}^{\prime} = A\mathbf{x}\) is given by the set of all solutions of the form \(\mathbf{x}_i(t) = e^{\lambda_i t} \mathbf{v}_i\). Each solution involves an exponential term with the eigenvalue and the corresponding eigenvector.

Solve the Initial Value Problem

We need to satisfy the initial condition \(\mathbf{x}(0) = \mathbf{x}_0\). Assume the solution takes the form \(\mathbf{x}(t) = c_1 \mathbf{x}_1(t) + c_2 \mathbf{x}_2(t) + c_3 \mathbf{x}_3(t) + c_4 \mathbf{x}_4(t)\). At \(t=0\), solve \(c_1 \mathbf{x}_1(0) + c_2 \mathbf{x}_2(0) + c_3 \mathbf{x}_3(0) + c_4 \mathbf{x}_4(0) = \mathbf{x}_0\) to find constants \(c_1, c_2, c_3, c_4\).

Write General Solution

Substitute the constants \(c_1, c_2, c_3, c_4\) back into the solution form \(\mathbf{x}(t) = c_1 \mathbf{x}_1(t) + c_2 \mathbf{x}_2(t) + c_3 \mathbf{x}_3(t) + c_4 \mathbf{x}_4(t)\) to obtain the specific solution that satisfies the initial condition.

Key concepts

Eigenvalues

Eigenvalues are crucial in understanding the behavior of linear systems, such as the one described by the matrix \(A\). They are essentially solutions to the characteristic equation \(\det(\lambda I - A) = 0\), where \(\lambda\) represents the eigenvalue. This determinant equation captures how a transformation associated with the matrix changes with respect to a scale factor. Here’s why they matter:

• **Predicting System Behavior:** Eigenvalues help determine whether the system grows, shrinks, or oscillates over time.
• **Related to Stability:** If all eigenvalues have negative real parts, the system tends towards stability.
• **Insight into Dynamics:** Different eigenvalues can reveal modes of oscillation or exponential growth/decay.

Understanding eigenvalues is your first step in decoding the dynamics of differential equations.

Eigenvectors

Once we have eigenvalues, corresponding eigenvectors are next. For an eigenvalue \(\lambda_i\), we find an eigenvector by solving \((A - \lambda_i I)\mathbf{v} = 0\). This equation searches for non-zero vectors \(\mathbf{v}\), which when transformed by \(A\), only scale by \(\lambda_i\) and don’t change direction.

• **Direction of Transformation:** Eigenvectors show intrinsic directions or patterns that remain consistent under the operation defined by matrix \(A\).
• **Dimensionality Role:** They help describe the geometry of the system that \(A\) governs, showing axes along which scaling occurs.
• **Essence of the Solution:** In solving ordinary differential equations, eigenvectors shape the form of solutions.

Eigenvectors are essential to understand how systems are oriented in space and contribute to forming particular solutions.

Initial Value Problem

An initial value problem (IVP) specifies conditions that a solution to a differential equation must satisfy at the start. It is represented as \(\mathbf{x}(0) = \mathbf{x}_0\), stating how the system behaves initially.

• **Provides Specificity:** While differential equations might have general solutions, an IVP narrows this down, identifying a single specific solution.
• **Initial Conditions Relevance:** These conditions set the unique trajectory of the system, crucial for real-world applications.
• **Essential for System Dynamics:** Solving an IVP means integrating the equation and applying the initial condition to find constants that personalize the general solution.

Understanding IVPs helps demonstrate how general solutions adapt to initial parameters to form a unique solution.

Fundamental Set of Solutions

A fundamental set of solutions to an ordinary differential equation forms the basic building blocks of all possible solutions. It generally involves finding multiplied solutions of the type \(\mathbf{x}_i(t) = e^{\lambda_i t} \mathbf{v}_i\), where \(\lambda_i\) and \(\mathbf{v}_i\) are the respective eigenvalue and eigenvector.

• **Complete System of Solutions:** This set allows construction of any solution for the system \(\mathbf{x}' = A\mathbf{x}\).
• **Combination of Solutions:** We find the full solution by combining the fundamental solutions with specific constants.
• **Tailored for Systems of Differential Equations:** Critical in applications that require multiple interconnected equations working simultaneously.

The fundamental set is like a toolkit, providing the essential functions needed to construct precise solutions to fit a given problem's unique demands.