Problem 11 - 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 \(A .\) \(\lambda_{1}=-1, \quad \lambda_{2}=-2 ; \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{r}{-1} \\ {1}\end{array}\right]\)

Step-by-Step Solution

Answer

The phase plane shows trajectories bending towards the origin, aligning with the negative eigenvalues.

Step 1: Understand the System of Differential Equations

The system given is \( \frac{d \mathbf{x}}{dt} = A \mathbf{x} \) where \( A \) is a matrix with the provided eigenvalues and eigenvectors. This means we will have solutions associated with each eigenvalue and eigenvector.

Step 2: Identify Solution Characteristics from Eigenvalues

Since both \( \lambda_1 = -1 \) and \( \lambda_2 = -2 \) are negative, the solutions to the differential equations are stable and will approach the origin over time. The trajectory along each eigenvector will determine the direction.

Step 3: Construct General Solution Using Eigenvectors

The general solution for the system is \( \mathbf{x}(t) = c_1 e^{-t} \begin{bmatrix} 1 \ 1 \end{bmatrix} + c_2 e^{-2t} \begin{bmatrix} -1 \ 1 \end{bmatrix} \). Here, \( c_1 \) and \( c_2 \) are constants determined by initial conditions, and \( e^{-t} \) and \( e^{-2t} \) show decay rates.

Step 4: Sketch Eigenvectors in the Phase Plane

Plot the eigenvectors in the phase plane: \(\begin{bmatrix} 1 \ 1 \end{bmatrix}\) pointing in direction \( (1,1) \) along the line \( y = x \) and \(\begin{bmatrix} -1 \ 1 \end{bmatrix}\) pointing in direction \( (-1,1) \) along the line \( y = -x \).

Step 5: Sketch Solution Curves

Solution curves approach the origin along the eigenvectors due to eigenvalue negativity. Initial points not on eigenvectors move along paths that curve towards the origin, bending towards the more dominant eigenvector at larger times.

Key Concepts

Differential Equations

A differential equation is a mathematical equation that relates a function with its derivatives. In the context of this exercise, we are dealing with a system of linear differential equations. These equations describe how a vector function changes with respect to time. The form given in this exercise is \( \frac{d \mathbf{x}}{dt} = A \mathbf{x} \), where \( A \) is a square matrix. This matrix encodes important information about the system's behavior. Each solution to the differential equation represents a path or trajectory that the system follows in its phase plane over time. Understanding these paths helps us predict the system's future behavior.

Eigenvalues

Eigenvalues are critical in determining the behavior of a differential equation system. They represent the growth rates of the solution. In our exercise, the eigenvalues given are \( \lambda_{1} = -1 \) and \( \lambda_{2} = -2 \). A negative eigenvalue, like those we have, indicates that the system's solutions diminish over time, leading the trajectories towards the origin. This is also referred to as stability. The larger the absolute value of the eigenvalue, the faster the decay rate. Here, \( \lambda_{2} = -2 \) suggests a faster decay in its associated direction compared to \( \lambda_{1} = -1 \). It is crucial to note that the stability of solutions is directly linked to these eigenvalues.

Eigenvectors

Eigenvectors give direction to the solution curves of a system of differential equations. They define invariant lines in the phase plane along which solutions approach or move away from the origin. In our problem, we have eigenvectors \( \mathbf{v}_{1} = \begin{bmatrix} 1 \ 1 \end{bmatrix} \) and \( \mathbf{v}_{2} = \begin{bmatrix} -1 \ 1 \end{bmatrix} \). These vectors point in the direction of the lines \( y = x \) and \( y = -x \) respectively. Each eigenvector is associated with an eigenvalue, dictating how solutions evolve along these directions. By plotting these eigenvectors, we can draw solution trajectories that curve towards the origin along these lines. This information helps visualize how the system behavior unfolds over time.

Stability

In the context of differential equations, stability refers to the tendency of solutions to converge to an equilibrium point. For our system, the equilibrium is the origin. Since our eigenvalues are negative, the system is stable. This means any perturbation or initial condition leads to solution curves that eventually return to or approach the origin. The negative eigenvalues ensure that as time progresses, solutions decay exponentially, showing stabilization. This stability shows how small deviations from equilibrium die out over time, with solutions moving along paths shaped by the eigenvectors.

General Solution

The general solution of a system of differential equations like \( \frac{d \mathbf{x}}{dt} = A \mathbf{x} \) provides a complete description of all possible states the system can achieve. It is formed by a combination of the system's eigenvectors, each multiplied by an exponential function of time that includes its corresponding eigenvalue. For our specific problem, the general solution is \( \mathbf{x}(t) = c_1 e^{-t} \begin{bmatrix} 1 \ 1 \end{bmatrix} + c_2 e^{-2t} \begin{bmatrix} -1 \ 1 \end{bmatrix} \). Here, \( c_1 \) and \( c_2 \) are constants based on initial conditions and \( e^{-t} \), \( e^{-2t} \) indicate how rapidly solutions decay over time. This blend of components ensures that any initial state will evolve in a predictable manner, allowing us to sketch phase plane trajectories effectively.