Question

Solve the given system of equations in each of Problems 20 through 23. Assume that \(t>0 .\) $$ t \mathbf{x}^{\prime}=\left(\begin{array}{rr}{5} & {-1} \\ {3} & {1}\end{array}\right) \mathbf{x} $$

Short answer

Answer: To solve this system for \(t > 0\), follow these steps: 1. Find the eigenvalues and eigenvectors of the matrix \(A\). 2. Rewrite the system in terms of the eigenvalues and eigenvectors. 3. Solve the system by separating variables and integrating both sides. 4. Combine the solutions found in step 3 to find the general solution \(\mathbf{x}(t)\).

Step-by-step solution

Find the eigenvalues and eigenvectors of A

To find the eigenvalues and eigenvectors of the matrix \(A\), we first determine the characteristic equation \((A - \lambda I)\mathrm{det} = 0\), where \(\lambda\) is an eigenvalue and \(I\) is the identity matrix. This gives us the equation: \( \begin{vmatrix} 5-\lambda & -1 \\ 3 & 1-\lambda \end{vmatrix} = (5-\lambda)(1-\lambda) - (-1)(3) = 0.\) Solve this equation for \(\lambda\) to obtain the eigenvalues. Next, use the eigenvalues to find the corresponding eigenvectors by solving \((A - \lambda I)\mathbf{v} = 0\) for each eigenvalue.

Rewrite the system in terms of the eigenvalues and eigenvectors

Substitute the eigenvectors found in step 1 into the given system of equations and rewrite the system using the eigenvalues and eigenvectors.

Solve the system by separating variables and integrating both sides

With the system rewritten in terms of eigenvalues and eigenvectors, we can now separate variables in the first-order differential equations and integrate both sides to find the solutions.

Combine the solutions to find the general solution \(\mathbf{x}(t)\)

Finally, using the solutions found in step 3, combine them to find the general solution for \(\mathbf{x}(t)\). This general solution depends on \(t\) and satisfies the given system of equations for all \(t > 0\).

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is crucial when solving systems of differential equations. In our case, we have a matrix A, which we can perceive as a transformation in space. An eigenvector, \( \mathbf{v} \), is a non-zero vector that changes by only a scalar factor when that transformation is applied. The scalar is known as the eigenvalue, \( \lambda \).

When we apply matrix A to eigenvector \( \mathbf{v} \), the vector \( \mathbf{v} \) remains in the same direction, and is only scaled by \( \lambda \). This property is expressed mathematically as A\mathbf{v}=\lambda\mathbf{v}. It's a fundamental concept because it allows us to decompose the system into simpler parts where each part corresponds to a single eigenvalue and eigenvector pair.

Characteristic Equation

Finding eigenvalues requires we set up and solve the characteristic equation, a determinant equation in the form (A - \lambda I)\mathrm{det} = 0, where I is the identity matrix and \lambda stands for an eigenvalue. This equation arises from the fact that for an eigenvector \( \mathbf{v} \) to exist, the system (A - \lambda I)\mathbf{v}=0 must be solvable.

For a 2x2 matrix as in the given exercise, the characteristic equation is a quadratic that can be factored or solved using the quadratic formula to determine the eigenvalues. These eigenvalues are crucial as they play a role in describing the behavior of the system over time.

First-order Differential Equations

First-order differential equations involve derivatives with respect to one variable, and they describe a wide variety of phenomena in engineering and physical sciences. In the context of systems of equations, we're often dealing with first-order linear differential equations, which can be solved using standard mathematical techniques.

Once a system is expressed in terms of eigenvalues and eigenvectors, each resultant first-order equation describes the evolution of the components of the system along the direction of its corresponding eigenvector. Solutions to these equations give us insight into the dynamics of the system for each separate mode.

Separation of Variables

Separation of variables is a technique used to solve differential equations, where the equation is re-arranged so that each variable is on a different side of the equation. This method can only be used when all the 'y' terms can be moved to one side of the equation and all the 'x' terms to the other side.

In practice, we integrate both sides—it's like 'unwrapping' the derivatives to get back to the original functions. For systems of differential equations, this method simplifies finding solutions that correspond to each eigenvalue-eigenvector pair by decoupling the system into independent equations.

General Solution for Differential Equations

The final goal when solving a system of differential equations is to find the general solution, a formula that captures all possible behaviors of the system under study. The general solution is built from the solutions to the individual first-order equations found using separation of variables and takes into account the initial conditions of the problem.

In our exercise, the general solution \(\mathbf{x}(t)\) incorporates the eigenvalues and eigenvectors, showing how the system evolves over time considering each mode of behavior. This is a composite solution that satisfies the entire system for any time t > 0, hence giving us a comprehensive understanding of the dynamics involved.