Problem 23 - Biocalculus Calculus for the Life Sciences

Question

Consider the system of linear differential equations \(d \mathbf{x} / d t=A \mathbf{x}+\mathbf{g},\) where \(\mathbf{g}\) is a vector of constants. Suppose that \(A\) is nonsingular. (a) What is the equilibrium of this system of equations? (b) Using \(\hat{\mathbf{x}}\) denote the equilibrium found in part \((a)\) define a new vector of variables \(\mathbf{y}=\mathbf{x}-\hat{\mathbf{x}} .\) What do the components of y represent? (c) Show that \(y\) satisfies the differential equation \(d \mathbf{y} / d t=A \mathbf{y} .\) This demonstrates how we can reduce a nonhomogeneous system of linear differential equations to a system that is homogenous by using a change of variables.

Step-by-Step Solution

Answer

(a) \( \hat{\mathbf{x}} = -A^{-1} \mathbf{g} \); (b) \( \mathbf{y} \) represents the deviation from equilibrium; (c) \( \mathbf{y} \) satisfies \( \frac{d \mathbf{y}}{d t} = A \mathbf{y} \).

Step 1: Write the equilibrium condition

The equilibrium of a system of differential equations is found where the time derivative is zero, i.e., \( \frac{d \mathbf{x}}{d t} = \mathbf{0} \). Hence, we need to solve the equation \( \mathbf{0} = A \mathbf{x} + \mathbf{g} \).

Step 2: Solve the equilibrium equation

Rearrange the equation to solve for the equilibrium state \( \hat{\mathbf{x}} \). Since \( A \) is nonsingular, the matrix has an inverse, allowing us to write:\[ A \hat{\mathbf{x}} = -\mathbf{g} \]Thus,\[ \hat{\mathbf{x}} = -A^{-1} \mathbf{g} \]This is the equilibrium solution of the system.

Step 3: Define the new variable \( \mathbf{y} \)

Define \( \mathbf{y} = \mathbf{x} - \hat{\mathbf{x}} \) where \( \hat{\mathbf{x}} \) is the equilibrium solution found earlier. The components of \( \mathbf{y} \) represent the deviation of \( \mathbf{x} \) from its equilibrium point.

Step 4: Differentiate \( \mathbf{y} \) with respect to time

Use the definition \( \mathbf{y} = \mathbf{x} - \hat{\mathbf{x}} \), and differentiate both sides with respect to time:\[ \frac{d \mathbf{y}}{d t} = \frac{d \mathbf{x}}{d t} - \frac{d \hat{\mathbf{x}}}{d t} \]Since \( \hat{\mathbf{x}} \) is constant, \( \frac{d \hat{\mathbf{x}}}{d t} = \mathbf{0} \), hence:\[ \frac{d \mathbf{y}}{d t} = \frac{d \mathbf{x}}{d t} \]

Step 5: Substitute into the original differential equation

From the given system \( \frac{d \mathbf{x}}{d t} = A \mathbf{x} + \mathbf{g} \) and substituting for \( \mathbf{x} = \mathbf{y} + \hat{\mathbf{x}} \), we have:\[ \frac{d \mathbf{y}}{d t} = A(\mathbf{y} + \hat{\mathbf{x}}) + \mathbf{g} \]Incorporating \( A \hat{\mathbf{x}} = -\mathbf{g} \), simplify the expression:\[ \frac{d \mathbf{y}}{d t} = A \mathbf{y} + A \hat{\mathbf{x}} + \mathbf{g} = A \mathbf{y} + \mathbf{0} = A \mathbf{y} \]

Step 6: Conclude the change of variables

Thus, \( \mathbf{y} \) satisfies the homogeneous differential equation \( \frac{d \mathbf{y}}{d t} = A \mathbf{y} \), demonstrating how the original nonhomogeneous system is transformed into a homogeneous system using a change of variables.

Key Concepts

Equilibrium Point

In the context of linear differential equations, an equilibrium point is where the system tends to settle and stop changing over time. This condition occurs when the rate of change or the derivative of the system is zero. For a system given by \(\frac{d \mathbf{x}}{dt} = A \mathbf{x} + \mathbf{g}\), the equilibrium is reached when \(\frac{d \mathbf{x}}{dt} = \mathbf{0}\).
This equilibrium condition simplifies to solving the equation \(A \mathbf{x} + \mathbf{g} = \mathbf{0}\).
Since \(A\) is a nonsingular matrix with an inverse, it allows us to rearrange this to find \(\hat{\mathbf{x}} = -A^{-1} \mathbf{g}\).

The equilibrium point \(\hat{\mathbf{x}}\) gives us the state of the system where it remains at rest, meaning all forces or inputs balance out. The existence and solvability of the equilibrium point rely heavily on the nonsingularity of the matrix \(A\), ensuring that it can be inverted.

Nonsingular Matrix

A matrix is termed nonsingular if it has an inverse, which is a key property in solving systems of equations. In a system like \(A \mathbf{x} = \mathbf{b}\), to solve for \(\mathbf{x}\), \(A\) must be nonsingular.

A nonsingular matrix is also referred to as an invertible or regular matrix. This property is crucial because it ensures that the system of equations doesn't collapse to a form where no unique solutions exist.
For example:

• The determinant of a nonsingular matrix is non-zero, a necessary condition for invertibility.
• Nonsingularity confirms that the matrix can be used to leverage \(A^{-1}\), the inverse, to isolate variables and solve the equations distinctly.
• This inverse property allows us to compute the equilibrium as \( \hat{\mathbf{x}} = -A^{-1} \mathbf{g} \), crucial for state-linearization in differential equations.

Thus, nonsingular matrices are foundational for ensuring that linear algebraic methods can be effectively applied to differential systems.

Change of Variables

In differential equations, transforming or simplifying the system is often achieved through a change of variables. Here, with \( \hat{\mathbf{x}} \) being the equilibrium, the new variable is defined as \( \mathbf{y} = \mathbf{x} - \hat{\mathbf{x}} \).
This transformation serves to measure the deviation of \( \mathbf{x} \) from its steady state.

Advantages of this approach include:

• It simplifies the analysis by shifting the perspective to how far the system is from equilibrium, rather than dealing with absolute values of \( \mathbf{x} \).
• The differential equation \( \frac{d \mathbf{y}}{dt} = A \mathbf{y} \) arises, indicating that the transformed system now behaves homogeneously.
• The process effectively helps in reducing complex systems into simpler forms, often linear in nature, which are easier to analyze and solve.

Ultimately, the change of variables is a powerful technique to make non-homogeneous systems approachable by converting them to homogeneous forms.

Homogeneous System

A homogeneous system of differential equations is one in which all terms are a function of the dependent variable and its derivatives, without any constant term added. In mathematical terms, for a system \( \frac{d \mathbf{y}}{dt} = A \mathbf{y} \), no constant vector \(\mathbf{g}\) appears.

The importance of homogeneous systems lies in their simplicity:

• They're easier to analyze due to their uniform structure.
• Solutions often involve exponentials of the matrix \(A\), resulting in straightforward characterizations of system behavior over time.
• With the absence of constant terms, tools like the matrix-exponential \(e^{At}\) become applicable for explicit solution finding.

This shift primarily occurs thanks to the change of variables, where a nonhomogeneous system is re-cast as a homogeneous one by factoring out the equilibrium state, thereby focusing solely on dynamic deviations \(\mathbf{y}\). This structural simplicity lends itself to robust mathematical treatment and analysis.