Problem 32 - Biocalculus Calculus for the Life Sciences

Question

A slightly more complicated system with repeated eigenvalues is $$\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{rr}{-1} & {1} \\ {0} & {-1}\end{array}\right] \mathbf{x}$$ (a) Show that \(x_{1}(t)=c_{1} e^{-t}+c_{2} t e^{-t}\) and \(x_{2}(t)=c_{2} e^{-t}\) is a solution. The origin in this case is called an improper node. (b) Try obtaining this general solution by calculating the eigenvectors and eigenvalues of the coefficient matrix. Comment on anything unusual that occurs.

Step-by-Step Solution

Answer

The original solution is verified, involving repeated eigenvalues leading to an improper node.

Step 1: Identify the Matrix

Start by identifying the matrix from the differential equation. This matrix is: \[A = \begin{pmatrix} -1 & 1 \ 0 & -1 \end{pmatrix}\]

Step 2: Find Eigenvalues

Find the eigenvalues of the matrix \(A\) by solving the characteristic equation: \(\det(A - \lambda I) = 0\), where \(I\) is the identity matrix.\[\begin{vmatrix} -1 - \lambda & 1 \ 0 & -1 - \lambda \end{vmatrix} = (-1 - \lambda)(-1 - \lambda) - 0 \cdot 1 = (\lambda + 1)^2 = 0\]This gives a repeated eigenvalue: \(\lambda = -1\).

Step 3: Calculate Eigenvectors

Find the eigenvector corresponding to the eigenvalue \(\lambda = -1\). Substitute \(\lambda = -1\) into \((A - \lambda I)\mathbf{v} = 0\):\[\begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \end{pmatrix}\]This results in \(v_1 = 0\). Choose \(v_2 = 1\), so the eigenvector is \(\mathbf{v} = \begin{pmatrix} 0 \ 1 \end{pmatrix}\). The second line provides no new condition, confirming the vector is indeed valid.

Step 4: Construct General Solution

Construct the general solution using the eigenvalue \(\lambda = -1\) and the eigenvector \(\mathbf{v}\). For repeated eigenvalues, an additional solution of the form \(\mathbf{x_2}(t) = \mathbf{v} t e^{-t}\) should be considered:\[\mathbf{x}(t) = c_1 \begin{pmatrix} 0 \ 1 \end{pmatrix} e^{-t} + c_2 \left(\begin{pmatrix} 0 \ 1 \end{pmatrix} t e^{-t} + \begin{pmatrix} 1 \ 0 \end{pmatrix} e^{-t}\right)\]

Step 5: Write Solution for Components

Write the solution in terms of the components \(x_1(t)\) and \(x_2(t)\): Since \(\begin{pmatrix} 1 \ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \ 1 \end{pmatrix}\) introduce terms for the eigenvectors, plug in:\[\begin{align*}x_1(t) &= c_1 \cdot 0 \cdot e^{-t} + c_2(1 \cdot t e^{-t} + 0 \cdot e^{-t}) = c_2 t e^{-t},\x_2(t) &= c_1 \cdot 1 \cdot e^{-t} + c_2(0 \cdot t e^{-t} + 1 \cdot e^{-t}) = c_1 e^{-t} + c_2 e^{-t}.\end{align*}\]Reorder to match forms, validating \(x_1(t) = c_1 e^{-t} + c_2 t e^{-t}\), \(x_2(t) = c_2 e^{-t}\).

Step 6: Conclusion and Comments

The original solution given was correct. We observed repeated eigenvalues necessitating the special solution form. This leads to the structure of an improper node in the eigenvalue problem, where a line of critical points exists, and solutions spiral into these points without distinct separation of directions.

Key Concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts when dealing with differential equations and linear transformations. They help us understand the behavior of systems and matrices. Let's break these down simply:

• **Eigenvalues** are special numbers associated with a matrix that describe the scaling factor applied to eigenvectors when the matrix is multiplied by them. For a matrix \(A\), an eigenvalue \(\lambda\) satisfies \(A\mathbf{v} = \lambda\mathbf{v}\), where \(\mathbf{v}\) is a non-zero vector called the eigenvector.
• **Eigenvectors** are vectors that do not change their direction under the associated linear transformation, only their magnitude gets scaled by the corresponding eigenvalue.

In our problem, we derive these by solving the characteristic equation of the matrix given in the differential equation. Identifying these helps in constructing solutions for systems of differential equations.

Improper Node

An improper node is a specific type of equilibrium point found in the study of dynamical systems, like our differential equation problem. This occurs in the phase plane where:

• The system has repeated real eigenvalues.
• The matrix does not have enough linearly independent eigenvectors.

This results in trajectories that tend to spiral towards or away from the equilibrium point without distinct separation of directions. Improper nodes indicate a degenerate situation where additional solutions, involving terms such as \(t e^{\lambda t}\), must be considered to fully capture the system's behavior.

Characteristic Equation

The characteristic equation is a polynomial equation derived from a matrix. It is crucial for finding the eigenvalues of a matrix. Here’s a step-by-step approach:

• Start with the matrix \(A\) from the differential equation.
• Subtract \(\lambda I\) from \(A\), where \(I\) is the identity matrix.
• Compute the determinant of the resulting matrix \(A - \lambda I\).
• Equate the determinant to zero and solve for \(\lambda\).

The roots of this equation are the eigenvalues. In our exercise, the determinant equation \((\lambda + 1)^2 = 0\) shows that \(-1\) is a repeated eigenvalue, leading to our special solutions.

Repeated Eigenvalues

Repeated eigenvalues occur when a characteristic equation yields the same eigenvalue more than once. Here’s why this matters:

• With repeated eigenvalues, particularly with multiplicity greater than one, typical solution methods using only eigenvectors may not suffice. This necessitates additional terms, often involving \(t\), in the solution.
• If the algebraic multiplicity (number of times the eigenvalue appears) differs from the geometric multiplicity (number of linearly independent eigenvectors), the full system solution needs these extra terms for completeness and accuracy.
• Repeated eigenvalues can result in behaviors like improper nodes in the phase plane, where distinct directional flows are absent.

Our problem demonstrated a repeated eigenvalue of \(-1\), requiring us to construct a special solution form to accommodate this scenario.