Problem 23 - Biocalculus Calculus for the Life Sciences

Question

Solve the initial value problem \(d \mathbf{x} / d t=A \mathbf{x}\) with \(\mathbf{x}(0)=\mathbf{x}_{0} .\) \(A=\left[ \begin{array}{rr}{-1} & {2} \\ {-3} & {-1}\end{array}\right] \quad \mathbf{x}_{0}=\left[ \begin{array}{l}{2} \\ {0}\end{array}\right]\)

Step-by-Step Solution

Answer

The solution is \( \mathbf{x}(t) = 2 e^{-t} \begin{bmatrix} \cos(2t) \\ -1.5 \sin(2t) \end{bmatrix} \).

Step 1: Set Up the Differential Equation

Given the differential equation \( \frac{d \mathbf{x}}{dt} = A \mathbf{x} \) with the matrix \( A = \begin{bmatrix} -1 & 2 \ -3 & -1 \end{bmatrix} \), and the initial condition \( \mathbf{x}(0) = \begin{bmatrix} 2 \ 0 \end{bmatrix} \). To solve this problem, we need to find the solution \( \mathbf{x}(t) \) that satisfies the initial condition.

Step 2: Diagonalize the Matrix A

Compute the eigenvalues of \( A \) by solving \( \text{det}(A - \lambda I) = 0 \), where \( I \) is the identity matrix. This gives the characteristic equation \( (-1-\lambda)((-1)-\lambda) - (2)(-3) = 0 \), simplifying to \( \lambda^2 + 2\lambda + 7 = 0 \). Solving for \( \lambda \), we find the eigenvalues are complex: \( \lambda_1 = -1 + 2i \), \( \lambda_2 = -1 - 2i \).

Step 3: Construct the General Solution

The general solution for a system with complex eigenvalues \( \lambda = \alpha \pm \beta i \) is given by \[\mathbf{x}(t) = c_1 e^{\alpha t}\begin{bmatrix} \text{Re}(v) \ \text{Im}(v) \end{bmatrix} \cos(\beta t) + c_2 e^{\alpha t}\begin{bmatrix} -\text{Im}(v) \ \text{Re}(v) \end{bmatrix} \sin(\beta t), \] where \( v \) is the eigenvector.

Step 4: Find Eigenvectors

For \( \lambda_1 = -1 + 2i \), solve \((A - (-1 + 2i)I)\mathbf{v} = \mathbf{0} \) to get the eigenvector \( \mathbf{v}_1 = \begin{bmatrix} 1 \ -1.5i \end{bmatrix} \). For the real and imaginary parts, let the eigenvector be \( \mathbf{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix} + i \begin{bmatrix} 0 \ -1.5 \end{bmatrix}. \)

Step 5: Substitute Initial Conditions

Substitute \( \mathbf{x}(0) = \begin{bmatrix} 2 \ 0 \end{bmatrix} \) into the general solution, which simplifies to find \( c_1 = 2 \) and \( c_2 = 0 \).

Step 6: Final Formulate Solution

Substitute \( c_1 \) and \( c_2 \) back into the general solution expression. This simplifies to: \[ \mathbf{x}(t) = 2 e^{-t} \begin{bmatrix} \cos(2t) \ -1.5 \sin(2t) \end{bmatrix}. \] This represents the unique solution given the initial conditions.

Key Concepts

Eigenvalues and Eigenvectors

In the realm of differential equations, particularly when dealing with matrices, eigenvalues and eigenvectors play a crucial role. They simplify the process of solving systems of linear equations, which is commonly represented in matrix form. An **eigenvalue** is a scalar that identifies how much a corresponding eigenvector is stretched or shrunk during a linear transformation represented by a matrix. Eigenvectors, on the other hand, are vectors that remain in the same direction during such transformations, merely being scaled by the eigenvalue.

To find the eigenvalues, often a characteristic equation is derived from the determinant of matrix subtracted by a scalar multiple of the identity matrix, set to zero — essentially solving \({det}(A - \lambda I) = 0\). Eigenvalues provide insights into the stability and behavior of dynamical systems.

• Step 1: Formulate the characteristic equation.
• Step 2: Solve for roots, giving eigenvalues.

After determining eigenvalues, we proceed to find their linked eigenvectors by solving \( (A - \lambda I) \mathbf{v} = 0\). This process involves linear algebra techniques and expresses how each eigenvector is only modified in magnitude (not direction) by the linear transformation.

Complex Solutions

In certain cases, especially exciting ones, matrices yield **complex eigenvalues**. This happens when the characteristic equation results in complex numbers. Complex eigenvalues often come as conjugate pairs, such as \(\lambda_1 = -1 + 2i\) and \(\lambda_2 = -1 - 2i\). These complex numbers indicate oscillatory components in the solution, commonly linked to systems with rotational dynamics.

The general solution involving complex eigenvalues calls for the use of exponential forms and trigonometric functions, where the real part dictates the exponential decay or growth, and the imaginary part incorporates sinusoidal behavior:

• The real part, \( e^{\alpha t} \), affects growth/shrinkage.
• The imaginary component, \( \beta \), interacts with \( \cos(\beta t) \) and \( \sin(\beta t) \) to introduce periodic changes.

In our solution, eigenvalues \(-1 \pm 2i\) lead to exponential \(e^{-t}\) and trigonometric functions \(\cos(2t)\) and \(\sin(2t)\), ripe for representing oscillatory motion seen in many physical systems.

Initial Value Problem

This technique focuses on finding a solution that satisfies both the differential equation and the given conditions at the starting point, or "initial conditions." In practice, an **initial value problem** involves solving for coefficients like \(c_1\) and \(c_2\) within the general solution to fit the initial state given by \( \mathbf{x}(0) \). It plays an essential role in uniquely determining the system's trajectory from a specific starting state.

For solving, substitute the initial condition into the general solution to solve for constants:

• Substitute the initial vector \(\mathbf{x}_0\) into \(\mathbf{x}(t)\).
• Solve resulting equations for the constants \(c_1\) and \(c_2\).

This process yields a precise solution that aligns with provided starting conditions, making it invaluable in prediction and control scenarios in fields such as physics, engineering, and economics.

Matrices in Calculus

**Matrices** form a central component in calculus, especially within systems of differential equations. They offer a compact form to represent and manipulate linear systems. In our problem, matrix \(A\) represents the set of coefficients driving changes in the vector \(\mathbf{x}(t)\). This approach aids simplification and solution of complex systems.

By harnessing the power of matrices, one can:

• Simplify representation of linear systems.
• Utilize matrix operations to determine solutions.
• Link system behavior to eigenproperties.

Calculus often extends into matrix algebra to manage more substantial, interconnected systems. The integration of matrices in solving differential equations exemplifies the versatility and power of mathematical tools, making them indispensable in scientific and engineering applications.