Problem 1 - Biocalculus Calculus for the Life Sciences

Question

Show that \(x_{1}(t)\) and \(x_{2}(t)\) are solutions to the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) . \(\begin{array}{l}{A=\left[ \begin{array}{cc}{3} & {-2} \\ {2} & {-2}\end{array}\right]} \\ {x_{1}(t)=\frac{1}{3}\left(4 e^{2 t}-e^{-t}\right),} & {x_{2}(t)=\frac{2}{3}\left(e^{2 t}-e^{-t}\right)}\end{array}\)

Step-by-Step Solution

Answer

Yes, \(x_1(t)\) and \(x_2(t)\) are indeed solutions to the differential equation \(d \mathbf{x} / d t= A \mathbf{x}\).

Step 1: Understanding the problem

We need to check if the given functions \(x_1(t)\) and \(x_2(t)\) are solutions to the system of differential equations, \(\frac{d \mathbf{x}}{dt} = A \mathbf{x}\). The system can be written as \(\frac{d}{dt}\begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 3 & -2 \ 2 & -2 \end{bmatrix}\begin{bmatrix} x_1 \ x_2 \end{bmatrix}\).

Step 2: Compute the derivatives

Calculate the derivatives \(\frac{d x_1}{dt}\) and \(\frac{d x_2}{dt}\). For \(x_1(t) = \frac{1}{3}(4e^{2t} - e^{-t})\), the derivative is \(x'_1(t) = \frac{1}{3}(8e^{2t} + e^{-t})\). For \(x_2(t) = \frac{2}{3}(e^{2t} - e^{-t})\), the derivative is \(x'_2(t) = \frac{2}{3}(2e^{2t} + e^{-t})\).

Step 3: Set up the matrix operation

Set up the matrix equation \(A\mathbf{x}\) for each component. Using the matrix \(A = \begin{bmatrix} 3 & -2 \ 2 & -2 \end{bmatrix}\), compute \(Ax\) using \(x_1(t)\) and \(x_2(t)\) as \(\mathbf{x}\). This gives us: \[A\begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix} = \begin{bmatrix} 3x_1(t) - 2x_2(t) \ 2x_1(t) - 2x_2(t) \end{bmatrix}\].

Step 4: Calculate \(A\mathbf{x}\)

Substitute \(x_1(t)\) and \(x_2(t)\) into the expression. Compute:\[3x_1(t) - 2x_2(t) = \frac{1}{3}(12e^{2t} - 3e^{-t}) - \frac{4}{3}(e^{2t} - e^{-t}) = \frac{1}{3}(8e^{2t} + e^{-t})\].\[2x_1(t) - 2x_2(t) = \frac{2}{3}(8e^{2t} - 2e^{-t}) - \frac{4}{3}(2e^{2t} - 2e^{-t}) = \frac{2}{3}(2e^{2t} + e^{-t})\].

Step 5: Verify equality

Compare the derivatives with the result from the matrix operation. We find that \[\begin{bmatrix} x'_1(t) \ x'_2(t) \end{bmatrix} = \begin{bmatrix} 3x_1(t) - 2x_2(t) \ 2x_1(t) - 2x_2(t) \end{bmatrix}\]. This confirms that the derivatives \(\frac{d}{dt}\begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix}\) are equal to \( A\begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix}\).

Key Concepts

Systems of Differential Equations

A system of differential equations is a set of equations that involves rates of change of multiple variables. These variables are typically dependent on a common independent variable, often time, denoted by \( t \). In our case, we have two equations involving the functions \( x_1(t) \) and \( x_2(t) \). Such systems are useful in modeling dynamic processes where several interdependent quantities change over time.
To solve a system like this, we need to find functions \( x_1(t) \) and \( x_2(t) \) that satisfy all equations in the system. Understanding these systems often requires techniques from calculus as well as an understanding of how these functions interact within the system.

Matrix Algebra

Matrix algebra provides a compact way to express systems of linear equations or differential equations. In our example, the system of equations is represented using matrix notation. We have the equation \( \frac{d \mathbf{x}}{dt} = A \mathbf{x} \), where \( A \) is a matrix and \( \mathbf{x} \) is a vector representing the functions \( x_1(t) \) and \( x_2(t) \).

• The matrix \( A \) organizes the coefficients that relate the equations to each other.
• The vector \( \mathbf{x} \) encapsulates multiple variables into a single entity for easier manipulation and computation.

Understanding matrix operations such as multiplication is crucial as it allows us to expand, solve, and verify systems of differential equations efficiently.

Solutions to Differential Equations

Finding solutions to differential equations means determining the function or functions that satisfy the given equations. For our system, solutions are given by the functions \( x_1(t) = \frac{1}{3}(4e^{2t} - e^{-t}) \) and \( x_2(t) = \frac{2}{3}(e^{2t} - e^{-t}) \). These functions are crafted to fulfill the requirements of the differential equations.
To validate if they are solutions:

• Derivatives (\( x'_1(t) \) and \( x'_2(t) \)) of the given functions must match the forms derived from the matrix equation \( A\mathbf{x} \).
• This involves taking the derivative of each function and substituting it back into the context of the given matrix operation.

When the original differential equations are satisfied by these solutions, it confirms they solve the system correctly.

Verification of Solutions

Verifying solutions involves proving that the computed solutions are consistent with the system of differential equations. This is essential to ensure the solutions correctly represent the behavior depicted by the system.
To verify, follow these steps:

• Calculate the derivative of each solution: \( x'_1(t) \) and \( x'_2(t) \).
• Substitute \( x_1(t) \) and \( x_2(t) \) into the matrix operation \( A\mathbf{x} \) and solve.
• Compare the resultant expressions from the above steps to the computed derivatives.

When you see that \( \begin{bmatrix} x'_1(t) & x'_2(t) \end{bmatrix} \) matches with the matrix product \( A\begin{bmatrix} x_1(t) & x_2(t) \end{bmatrix} \), you have confirmed the solutions are verified. This step is crucial to validate the mathematical integrity of the solutions in real-world applications.