Problem 4 - Biocalculus Calculus for the Life Sciences

Question

Show that \(x_{1}(t)\) and \(x_{2}(t)\) are solutions to the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) . \(\begin{array}{l}{A=\left[ \begin{array}{rr}{1} & {2} \\ {-4} & {1}\end{array}\right]} \\ {x_{1}(t)=e^{t} \cos (2 \sqrt{2} t),} & {x_{2}(t)=-\sqrt{2} e^{t} \sin (2 \sqrt{2} t)}\end{array}\)

Step-by-Step Solution

Answer

Both functions satisfy the system of differential equations; they are solutions.

Step 1: Understand the Problem

We are given two functions \(x_1(t)\) and \(x_2(t)\) and have to show that they are solutions to the differential equation \(d \mathbf{x}/dt = A \mathbf{x}\). The matrix \(A\) is \(\begin{bmatrix} 1 & 2 \ -4 & 1 \end{bmatrix}\). Our goal is to check if \(\mathbf{x}(t) = \begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix}\) satisfies the equation with this matrix \(A\).

Step 2: Compute Derivatives

First, we calculate the derivatives of the given functions: 1. \(x_1(t) = e^t \cos(2\sqrt{2}t)\), differentiate using the product rule: \( x_1'(t) = e^t \cos(2\sqrt{2}t) + e^t (-2\sqrt{2} \sin(2\sqrt{2}t)) \).2. \(x_2(t) = -\sqrt{2} e^t \sin(2\sqrt{2}t)\), differentiate using the product rule: \( x_2'(t) = -\sqrt{2} e^t \sin(2\sqrt{2}t) + -\sqrt{2} e^t (2\sqrt{2}\cos(2\sqrt{2}t)) \).

Step 3: Simplify the Derivatives

Upon simplifying, we find:1. For \(x_1'(t)\): \[ x_1'(t) = e^t (\cos(2\sqrt{2}t) - 2\sqrt{2} \sin(2\sqrt{2}t)) \].2. For \(x_2'(t)\): \[ x_2'(t) = -\sqrt{2}e^t (\sin(2\sqrt{2}t) + 2\sqrt{2} \cos(2\sqrt{2}t)) \].

Step 4: Confirm the Matrix Equation

Substitute \(x_1(t)\) and \(x_2(t)\) into the matrix equation \[d \mathbf{x} / dt = A \mathbf{x} = \begin{bmatrix} 1 & 2 \ -4 & 1 \end{bmatrix} \begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix}\].Calculate:- The right-hand side gives \[ \begin{bmatrix} 1 & 2 \ -4 & 1 \end{bmatrix} \begin{bmatrix} e^t \cos(2\sqrt{2}t) \ -\sqrt{2} e^t \sin(2\sqrt{2}t) \end{bmatrix} = \begin{bmatrix} e^t \cos(2\sqrt{2}t) - 2\sqrt{2} e^t \sin(2\sqrt{2}t) \ -4e^t \cos(2\sqrt{2}t) -\sqrt{2} e^t \sin(2\sqrt{2}t) \end{bmatrix} \].The results match the simplified derivatives obtained in Step 3, confirming \(x_1(t)\) and \(x_2(t)\) satisfy the differential equation.

Key Concepts

Differential Equations

Differential equations are a type of equation that involve a function and its derivatives. They are essential in modeling various real-world phenomena such as heat, sound, electrodynamics, and much more. A differential equation ties functions and their rates of change in a single representation, allowing us to understand how systems evolve over time.

Understanding differential equations involves recognizing the type of equation, such as ordinary or partial, and the degree of the derivative involved. The overall goal is to find a function or a set of functions that satisfy the relation expressed by the equation.

• Ordinary Differential Equations (ODEs): These involve functions of a single variable and their derivatives.
• Partial Differential Equations (PDEs): These are more complex and involve derivatives of functions of multiple variables.

Solving these equations either exactly or approximately allows us to predict the behavior of dynamic systems under various conditions.

Matrix Differential Equations

Matrix differential equations extend the concept of differential equations to higher dimensions by using matrices to represent linear systems and their interrelated variables. This approach is especially useful in fields such as engineering, physics, and applied mathematics, where systems are described by multiple equations.

In a matrix differential equation, the state of a system is represented as a vector, and the system's evolution over time is described by how this vector changes, governed by matrix multiplication. The equation can be written as \( \frac{d\mathbf{x}}{dt} = A\mathbf{x} \), where:

• \( \mathbf{x} \) is a vector of functions representing system states.
• \( A \) is a matrix containing coefficients that determine how these states influence each other.

The process of finding solutions involves determining a set of functions that satisfy this matrix equation, giving insight into how variables within the system evolve simultaneously.

Solutions Verification

Solutions verification is a critical step in solving differential equations, especially matrix differential equations. It involves checking whether a proposed solution satisfies the original differential equation. This step ensures no errors were made during the solving process and confirms the accuracy and validity of the solution.

Verification typically involves three steps:

• Derive the solutions' derivatives by following mathematical rules such as the product rule or chain rule.
• Plug these derivatives back into the original differential equation to check consistency.
• Compare both sides of the equation - they should match if the functions are a correct solution.

This process helps avoid misinterpretations and mathematical mishaps, confirming that the selected solutions accurately describe the modeled system.

Linear Algebra

Linear algebra provides the foundational tools necessary to handle matrix differential equations. It involves the study of vectors, matrices, and linear transformations. Understanding these concepts enables us to manipulate complex systems efficiently.

Key linear algebra concepts relevant to differential equations include:

• Matrix Operations: Allows for the manipulation of equations and understanding how different states interact within a system.
• Eigenvectors and Eigenvalues: Offer insights into the behavior and stability of differential systems by describing directions of scaling and rotation.
• Determinants and Inverses: Provide methods to solve for unknown variables and ensure transformations are valid.

By applying these concepts, one can intelligently interpret and solve matrix differential equations, turning abstract mathematical models into tangible insights about real-world phenomena.