Problem 16 - Calculus for Biology and Medicine

Question

Find the general solution of each given system of differential equations and sketch the lines in the direction of the eigenvectors. Indicate on each line the direction in which the solution would move if it starts on that line. $$ \left[\begin{array}{c} \frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t} \end{array}\right]=\left[\begin{array}{ll} -5 & 3 \\ -2 & 0 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$

Step-by-Step Solution

Answer

The general solution is \( \mathbf{x}(t) = c_1 \begin{bmatrix} 1 \\ 1 \end{bmatrix}e^{-2t} + c_2 \begin{bmatrix} 3 \\ 2 \end{bmatrix}e^{-3t} \) with flow towards the origin.

Step 1: Write Down the System of Differential Equations

The given system is expressed as: \( \frac{d\mathbf{x}}{dt} = A \mathbf{x} \), where \( A = \begin{bmatrix} -5 & 3 \ -2 & 0 \end{bmatrix} \) and \( \mathbf{x} = \begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix} \). Our task is to find the general solution of this system.

Step 2: Find the Eigenvalues of the Matrix A

Calculate the eigenvalues \( \lambda \) from the characteristic equation \( \det(A - \lambda I) = 0 \).\[\det\left(\begin{bmatrix} -5-\lambda & 3 \ -2 & -\lambda \end{bmatrix}\right) = (\lambda + 5)(\lambda) + 6 = \lambda^2 + 5\lambda + 6.\] Solving \( \lambda^2 + 5\lambda + 6 = 0 \) gives \( \lambda_1 = -2 \) and \( \lambda_2 = -3 \).

Step 3: Find the Eigenvectors for Each Eigenvalue

For \( \lambda_1 = -2 \), solve \((A - (-2)I)\mathbf{v_1} = 0\).\[\begin{bmatrix} -3 & 3 \ -2 & 2 \end{bmatrix}\begin{bmatrix} v_{11} \ v_{12} \end{bmatrix}=\begin{bmatrix} 0 \ 0 \end{bmatrix}.\] This simplifies to \(-3v_{11} + 3v_{12} = 0\) which gives \( v_{12} = v_{11} \). A possible eigenvector is \( \mathbf{v_1} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \).For \( \lambda_2 = -3 \), solve \((A - (-3)I)\mathbf{v_2} = 0\).\[\begin{bmatrix} -2 & 3 \ -2 & 3 \end{bmatrix}\begin{bmatrix} v_{21} \ v_{22} \end{bmatrix}=\begin{bmatrix} 0 \ 0 \end{bmatrix}.\] This simplifies to \(-2v_{21} + 3v_{22} = 0\) which gives \( v_{22} = \frac{2}{3}v_{21} \). A possible eigenvector is \( \mathbf{v_2} = \begin{bmatrix} 3 \ 2 \end{bmatrix} \).

Step 4: Write Down the General Solution

Using the eigenvalues and eigenvectors, the general solution can be written as:\[ \mathbf{x}(t) = c_1 \begin{bmatrix} 1 \ 1 \end{bmatrix}e^{-2t} + c_2 \begin{bmatrix} 3 \ 2 \end{bmatrix}e^{-3t}, \]where \( c_1 \) and \( c_2 \) are constants determined by initial conditions.

Step 5: Analyze the Direction of the Eigenvectors

Since both eigenvalues are negative, the solutions \( e^{-2t} \) and \( e^{-3t} \) decay to zero as \( t \to \infty \). This means that the direction of the flow along the eigenvectors is towards the origin. The vector \( \begin{bmatrix} 1 \ 1 \end{bmatrix} \) represents the line \( x_1 = x_2 \), and the vector \( \begin{bmatrix} 3 \ 2 \end{bmatrix} \) represents the line \( x_2 = \frac{2}{3}x_1 \). Arrows point towards the origin on these lines.

Step 6: Sketch the Direction Field Using Eigenvectors

Sketch the lines corresponding to the eigenvectors on the phase plane. Indicate the direction of the flow with arrows pointing towards the origin along these lines: - For \( \begin{bmatrix} 1 \ 1 \end{bmatrix} \), draw a line through the origin with a slope of 1. - For \( \begin{bmatrix} 3 \ 2 \end{bmatrix} \), draw a line through the origin with a slope of \( \frac{2}{3} \).

Key Concepts

Eigenvalues

Eigenvalues are crucial in understanding systems of differential equations. They are scalars that give us insight into the system's nature by showing how the matrix (system) influences the direction of its vectors.
In your exercise, eigenvalues are determined by solving the characteristic equation: \( \det(A - \lambda I) = 0 \).From the matrix given, you get a quadratic equation: \( \lambda^2 + 5\lambda + 6 = 0 \).
By finding the roots of this equation, you discover the eigenvalues \( \lambda_1 = -2 \) and \( \lambda_2 = -3 \). Each of these values plays a significant role:

• Negative eigenvalues indicate that the system's solutions will move towards the origin, which means they decay over time.
• Eigenvalues help portray the solutions of the differential equation as a combination of exponential functions.

Understanding eigenvalues help predict the system's behavior, creating models that illustrate things like decay, stability, or growth.

Eigenvectors

Once eigenvalues are determined, we find the corresponding eigenvectors. These are non-zero vectors that reveal important geometric information about the system. They show the direction of the flow of solutions in the phase plane.
For each eigenvalue, solve \((A - \lambda I) \mathbf{v} = 0 \) to find the eigenvector.For example:

• For \( \lambda_1 = -2 \), the calculation yields an eigenvector \( \mathbf{v_1} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \). This indicates a direction, or line, where the phase plane solutions flow.
• For \( \lambda_2 = -3 \), solve to find \( \mathbf{v_2} = \begin{bmatrix} 3 \ 2 \end{bmatrix} \). This denotes another direction in which solutions tend to move.

These vectors allow you to sketch the solution's direction in a coordinate system. They reveal lines along which solutions decay or grow, depending on the nature of the eigenvalues. In this case, they all decay toward the origin since both eigenvalues are negative.

Matrix Analysis

Matrix analysis in differential equations allows us to solve systems like the one presented by systematically breaking down the characteristics of the matrix. This approach provides a structured way to handle systems using algebraic properties.
Begin by recognizing your system as \( \frac{d\mathbf{x}}{dt} = A \mathbf{x} \) with matrix \( A = \begin{bmatrix} -5 & 3 \ -2 & 0 \end{bmatrix} \).

• The matrix defines the system's dynamics, showing how different equations interact and influence each other.
• By conducting matrix operations, such as calculating determinants, we derive eigenvalues, which inform us about the system's behavior over time.
• Further, matrix operations help to find eigenvectors that guide us on sketching directions on the phase plane.

By combining these elements — the eigenvalues, eigenvectors, and matrix operations — you can accurately predict and visualize system responses. Such analysis not only solves differential equations but also enhances understanding of complex systems in engineering, physics, and beyond.