Problem 15 - Biocalculus Calculus for the Life Sciences

Question

Sketch several solution curves in the phase plane of the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) using the given eigenvalues and eigenvectors of \(A .\) \(\lambda_{1}=5, \quad \lambda_{2}=1 ; \quad \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{2} \\ {2}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{r}{-2} \\ {1}\end{array}\right]\)

Step-by-Step Solution

Answer

The phase plane shows an unstable node with trajectories moving away along directions defined by eigenvectors.

Step 1: Write the General Solution

Given the system \( \frac{d \mathbf{x}}{d t} = A \mathbf{x} \), we use the eigenvalues and eigenvectors to form the general solution. Since \( \lambda_1 = 5 \) and \( \lambda_2 = 1 \), the solution can be written as \( \mathbf{x}(t) = c_1 e^{5t} \mathbf{v}_1 + c_2 e^{t} \mathbf{v}_2 \), where \( c_1 \) and \( c_2 \) are constants, and \( \mathbf{v}_1 = \begin{bmatrix} 2 \ 2 \end{bmatrix} \) and \( \mathbf{v}_2 = \begin{bmatrix} -2 \ 1 \end{bmatrix} \).

Step 2: Analyze Eigenvalues for Stability and Direction

The eigenvalues \( \lambda_1 = 5 \) and \( \lambda_2 = 1 \) are both positive, indicating that solutions will move away from the origin over time; this is an unstable node. Because \( \lambda_1 > \lambda_2 \), trajectories will initially follow the direction of \( \mathbf{v}_1 \) more strongly.

Step 3: Determine General Directions

The direction associated with each eigenvector, \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \), helps establish the trajectory paths. Solutions near the origin will start in the direction of \( \mathbf{v}_1 = \begin{bmatrix} 2 \ 2 \end{bmatrix} \) and eventually follow a path more aligned with \( \mathbf{v}_2 = \begin{bmatrix} -2 \ 1 \end{bmatrix} \).

Step 4: Sketch the Phase Plane

Create the phase plane by plotting the eigenvectors as straight lines through the origin: \( \mathbf{v}_1 \) along \( x = y \) and \( \mathbf{v}_2 \) along \( 2x + y = 0 \). Draw trajectories starting along the direction of \( \mathbf{v}_1 \) and curving to become more aligned with \( \mathbf{v}_2 \). All trajectories move outwards, confirming instability.

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. They are essential in modeling various real-world phenomena where rates of change are crucial, such as physics, engineering, and biology. In this context, we are dealing with a **system of differential equations**, which involves more than one interrelated equation. These equations describe how variables change with respect to another variable, often time.

For the given problem, the differential equation is written as \( \frac{d \mathbf{x}}{dt} = A \mathbf{x} \). Here, \( \mathbf{x} \) is a vector function of time \( t \), and \( A \) is a constant matrix. The task is to solve this differential equation to understand how the system behaves over time. Solving such systems often involves linear algebraic methods because the behavior of the system is determined by the properties of matrix \( A \).

In particular, finding how the solutions evolve involves determining the eigenvalues and eigenvectors of \( A \). These will give us insights into the system's qualitative behavior and help us sketch the phase plane.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are foundational concepts in linear algebra, particularly useful for analyzing linear transformations represented by matrices. For a matrix \( A \), an eigenvector \( \mathbf{v} \) and corresponding eigenvalue \( \lambda \) satisfy the equation \( A\mathbf{v} = \lambda\mathbf{v} \).

In the given problem, we have two eigenvalues, \( \lambda_1 = 5 \) and \( \lambda_2 = 1 \), with corresponding eigenvectors \( \mathbf{v}_1 = \begin{bmatrix} 2 \ 2 \end{bmatrix} \) and \( \mathbf{v}_2 = \begin{bmatrix} -2 \ 1 \end{bmatrix} \). These values and vectors help us construct the general solution of the differential equation:

\[ \mathbf{x}(t) = c_1 e^{5t} \begin{bmatrix} 2 \ 2 \end{bmatrix} + c_2 e^{t} \begin{bmatrix} -2 \ 1 \end{bmatrix} \]

This expression shows that the solution \( \mathbf{x}(t) \) is a combination of exponential functions scaled by the eigenvectors. The exponential terms indicate how quickly solutions grow or decay as time progresses, determined by the eigenvalues. The constants \( c_1 \) and \( c_2 \) determine the specific path of a solution curve based on initial conditions. Together, they form a complete picture of the system's dynamics.

Stability Analysis

Stability analysis involves examining whether solutions to a differential equation system converge to an equilibrium point, stay constant, or diverge as time progresses. For linear systems like the one we're working with, stability is heavily influenced by the eigenvalues of matrix \( A \).

In this problem, both eigenvalues \( \lambda_1 = 5 \) and \( \lambda_2 = 1 \) are positive. This indicates that solutions will tend to move away from the equilibrium point at the origin, classifying the system as an **unstable node**. The fact that both eigenvalues are greater than zero is key: it means all trajectories in the phase plane spiral outward or "explode" from the origin as \( t \to \infty \).

Moreover, since \( \lambda_1 \) is greater than \( \lambda_2 \), solutions will initially follow the trajectory defined more strongly by the eigenvector \( \mathbf{v}_1 = \begin{bmatrix} 2 \ 2 \end{bmatrix} \), before aligning more closely with \( \mathbf{v}_2 = \begin{bmatrix} -2 \ 1 \end{bmatrix} \). This directionality tells us how the solution curves will progress in the phase plane, helping sketch them accurately. Such analysis is vital in predicting the long-term behavior of dynamic systems.