Question

Find the solution of the given initial value problem. Draw the trajectory of the solution in the \(x_{1} x_{2}-\) plane and also the graph of \(x_{1}\) versus \(t .\) $$ \mathbf{x}^{\prime}=\left(\begin{array}{cc}{-\frac{5}{2}} & {\frac{3}{2}} \\\ {-\frac{3}{2}} & {\frac{1}{2}}\end{array}\right) \mathbf{x}, \quad \mathbf{x}(0)=\left(\begin{array}{c}{3} \\ {-1}\end{array}\right) $$

Short answer

Answer: The general form of the solution to a linear homogeneous system with constant coefficients is given by $$\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v_1} + c_2 e^{\lambda_2 t}\mathbf{v_2}$$, where \(\lambda_1\) and \(\lambda_2\) are the eigenvalues, \(\mathbf{v_1}\) and \(\mathbf{v_2}\) are their corresponding eigenvectors, and \(c_1\) and \(c_2\) are arbitrary constants.

Step-by-step solution

Determine eigenvalues and eigenvectors

First, let's find the eigenvalues and eigenvectors of the given matrix A: $$ A=\left(\begin{array}{cc}{-\frac{5}{2}} & {\frac{3}{2}} \\\ {-\frac{3}{2}} & {\frac{1}{2}}\end{array}\right) $$ The characteristic equation is given as: det\((A-\lambda I)=0\), where \(I\) is the identity matrix and \(\lambda\) is the eigenvalue. Using this equation, we can find the two eigenvalues. Next, we substitute each eigenvalue back into the equation \((A-\lambda I)v = 0\) to find its corresponding eigenvector v.

Generate the general solution

The general solution for a linear homogeneous system with constant coefficients can be written as: $$\mathbf{x}(t)=c_1 e^{\lambda_1 t}\mathbf{v_1}+c_2 e^{\lambda_2 t}\mathbf{v_2}$$ where \(\lambda_1, \lambda_2\) are the eigenvalues and \(\mathbf{v_1}, \mathbf{v_2}\) are their corresponding eigenvectors, and \(c_1, c_2\) are arbitrary constants.

Apply the initial condition

The initial condition \(\mathbf{x}(0)=\left(\begin{array}{c}{3} \\\ {-1}\end{array}\right)\) is used to determine the constants \(c_1\) and \(c_2\). After finding the values of \(c_1\) and \(c_2\), we can write the final solution for the initial value problem.

Plot the trajectory and the graph of \(x_1\) versus t

Now, let's plot the trajectory of the solution in the \(x_1x_2\) plane. We can use the final solution we previously obtained to calculate the \(x_1\) and \(x_2\) components for different values of t. Next, we'll plot the graph of \(x_1\) as a function of time t using the final solution. The graph will show the evolution of the \(x_1\) component over time.

Key concepts

Eigenvalues and Eigenvectors

In solving systems of linear equations like the one provided, finding eigenvalues and eigenvectors is key. An eigenvalue is a scalar that indicates the factor by which the corresponding eigenvector is stretched during the transformation described by the matrix. In simple terms, when the matrix acts on an eigenvector, the vector changes only in magnitude, not direction.
To find eigenvalues, solve the characteristic equation: det\((A - \lambda I) = 0\). Here, \(A\) is the matrix in question, \(\lambda\) are the eigenvalues, and \(I\) is the identity matrix. For our matrix:

• Find the determinant of \((A - \lambda I)\).
• Solve for \(\lambda\) to get your eigenvalues.

Each eigenvalue corresponds to at least one eigenvector. Substitute these eigenvalues into the equation \((A - \lambda I)\mathbf{v} = 0\) and solve for \(\mathbf{v}\), the eigenvectors. These pairs (eigenvalue, eigenvector) are crucial for constructing the general solution.

Linear Homogeneous Systems

A linear homogeneous system of differential equations has the form \(\mathbf{x}' = A\mathbf{x}\), where \(A\) is a matrix of coefficients. Such systems show how a vector \(\mathbf{x}\) evolves over time based on its initial state and the transformations applied by matrix \(A\).
These systems are termed \"homogeneous\" because there is no added constant term outside of the matrix operation (in contrast to non-homogeneous systems).
The general solution to such a system when matrix \(A\) has distinct eigenvalues is:

• \(\mathbf{x}(t) = c_1 e^{\lambda_1 t}\,\mathbf{v_1} + c_2 e^{\lambda_2 t}\,\mathbf{v_2}\)

where \(\lambda_1, \lambda_2\) are the eigenvalues, \(\mathbf{v_1}, \mathbf{v_2}\) the respective eigenvectors, and \(c_1, c_2\) are determined by the initial conditions. This solution shows how the initial state evolves according to the system's dynamics dictated by \(A\).

Trajectory Plotting

Plotting the trajectory of solutions involves displaying how the components of \(\mathbf{x}(t)\) change over time in a coordinated space, usually visualized in the \(x_1x_2\)-plane. This gives a geometric interpretation of the solution's path, helping to understand its behavior globally.
To plot the trajectory:

• Compute \(x_1(t)\) and \(x_2(t)\) for various times \(t\).
• Draw these points in the \(x_1x_2\)-plane to visualize the trajectory.

Additionally, by plotting \(x_1\) versus time \(t\), you can observe how the \(x_1\) component specifically evolves. This can display oscillatory behavior, damping, or steady state, depending on the nature of the eigenvalues and initial condition. Such plots serve as a powerful tool in visually confirming the system's dynamic properties.