Problem 12 - Advanced Engineering Mathematics

Question

Discuss the geometric nature of the solutions to the linear system \(\mathbf{X}^{\prime}=\mathbf{A X}\) given the general solution. (a) \(\mathbf{X}(t)=c_{1}\left(\begin{array}{l}1 \\ 1\end{array}\right) e^{-t}+c_{2}\left(\begin{array}{r}1 \\ -2\end{array}\right) e^{-2 t}\) (b) \(\mathbf{X}(t)=c_{1}\left(\begin{array}{r}1 \\ -1\end{array}\right) e^{-t}+c_{2}\left(\begin{array}{l}1 \\ 2\end{array}\right) e^{2 t}\)

Step-by-Step Solution

Answer

Part (a) results in decay along eigenvectors showing a saddle point. Part (b) shows behavior typical of a saddle, with decay and growth directions.

Step 1: Analyze part (a)

The general solution for part (a) is given by \( \mathbf{X}(t)=c_{1}\left(\begin{array}{l}1 \ 1\end{array}\right) e^{-t}+c_{2}\left(\begin{array}{r}1 \ -2\end{array}\right) e^{-2 t} \).This solution consists of two terms, each representing a vector solution scaled by exponential functions. The terms \( e^{-t} \) and \( e^{-2t} \) suggest exponential decay.

Step 2: Identify the eigenvectors in part (a)

The vectors \( \begin{pmatrix} 1 \ 1 \end{pmatrix} \) and \( \begin{pmatrix} 1 \ -2 \end{pmatrix} \) are the eigenvectors associated with part (a). These vectors represent directions in the plane. The first vector \( \begin{pmatrix} 1 \ 1 \end{pmatrix} \) lies along the line \( x=y \), while the second vector \( \begin{pmatrix} 1 \ -2 \end{pmatrix} \) lies along the line \( x=-2y \).

Step 3: Geometric nature of solutions in part (a)

In part (a), the general solution describes a saddle point at the origin. The solutions decay towards the origin along the eigenvector directions as \( t \rightarrow \infty \), but do so at different rates because of the different exponential terms.

Step 4: Analyze part (b)

The general solution for part (b) is given by \( \mathbf{X}(t)=c_{1}\left(\begin{array}{r}1 \ -1\end{array}\right) e^{-t}+c_{2}\left(\begin{array}{l}1 \ 2\end{array}\right) e^{2 t} \).Here, the solution includes both a decaying \( e^{-t} \) and a growing \( e^{2t} \) exponential, leading to different stability properties.

Step 5: Identify the eigenvectors in part (b)

The vectors \( \begin{pmatrix} 1 \ -1 \end{pmatrix} \) and \( \begin{pmatrix} 1 \ 2 \end{pmatrix} \) are eigenvectors in part (b). The first vector \( \begin{pmatrix} 1 \ -1 \end{pmatrix} \) lies along the line \( x = -y \) and the second vector \( \begin{pmatrix} 1 \ 2 \end{pmatrix} \) lies along the line \( x=2y \).

Step 6: Geometric nature of solutions in part (b)

In part (b), the solutions exhibit a saddle point at the origin as well. However, solutions decay towards the origin along the direction of \( \begin{pmatrix} 1 \ -1 \end{pmatrix} \), and solutions grow away from the origin along \( \begin{pmatrix} 1 \ 2 \end{pmatrix} \), due to the respective exponential terms.

Key Concepts

Eigenvectors

When dealing with linear differential equations, eigenvectors play an essential role in understanding and interpreting the solutions. In the context of the given exercise, eigenvectors can be seen as directions in the plane along which the system moves over time.

For instance, in part (a) of the exercise, the eigenvectors \( \begin{pmatrix} 1 \ 1 \end{pmatrix} \) and \( \begin{pmatrix} 1 \ -2 \end{pmatrix} \) represent two distinct paths. The first path, or line, aligns with the equation \( x = y \), and the second aligns with \( x = -2y \).

Similarly, for part (b), the eigenvectors \( \begin{pmatrix} 1 \ -1 \end{pmatrix} \) and \( \begin{pmatrix} 1 \ 2 \end{pmatrix} \) describe the lines \( x = -y \) and \( x = 2y \) respectively.

Understanding these paths is crucial because they offer insight into the overall behavior of the system. Each eigenvector provides a direction along which the interplay of forces within the system is revealed. Knowing these directions helps predict how solutions to the differential equations will evolve over time.

Exponential Decay

Exponential decay is a phenomenon that describes how quantities reduce over time, often seen in decimal-based functions like \( e^{-t} \).

In the exercise, exponential decay appears prominently in the general solutions of part (a). Exponential functions such as \( e^{-t} \) and \( e^{-2t} \) illustrate how solutions to the differential equations decrease as time progresses. These functions modulate the influence of each eigenvector in the solution.

This decay occurs at different rates due to the various exponential terms. For example, \( e^{-t} \) implies a slower decay compared to \( e^{-2t} \). It means that the influence along the direction \( \begin{pmatrix} 1 \ 1 \end{pmatrix} \) diminishes slower than along \( \begin{pmatrix} 1 \ -2 \end{pmatrix} \). This discrepancy in rates impacts the dynamics of how the system behaves, particularly in how it approaches stability or equilibrium.

Understanding exponential decay is vital when analyzing linear differential equations, as it often governs the convergence of solutions towards equilibrium points or their dispersion away from them.

Saddle Points

Saddle points in the context of linear differential equations often indicate intersections in phase space where the behavior of solutions changes.

For both parts (a) and (b) of the exercise, the origin acts as a saddle point. This means the solutions behave in a specific way as they approach or move away from the origin in the plane defined by the eigenvectors.

In part (a), due to the presence of exponential decay, the system describes behavior moving towards the origin, which is a characteristic of a stable node. However, since different directions decay at varying rates, rather than a simple node, it's a saddle point.

In part (b), one direction described by \( e^{-t} \) shows decay, while the other, \( e^{2t} \), shows growth. This unique interplay indicates the existence of a saddle point with trajectories that are attracted along one eigenvector direction but repelled along another.

These saddle points are central to understanding system dynamics, as they highlight paths in the phase space where movements change direction, creating complex behaviors often found in systems exhibiting both stability and instability tendencies.