Problem 20 - Calculus for Biology and Medicine

Question

Solve the given initial-value problem. \(\left[\begin{array}{l}\frac{d x_{1}}{d t} \\ \frac{d x_{2}}{d t}\end{array}\right]=\left[\begin{array}{ll}1 & 3 \\ 0 & 2\end{array}\right]\left[\begin{array}{l}x_{1}(t) \\\ x_{2}(t)\end{array}\right]\) with \(x_{1}(0)=2\) and \(x_{2}(0)=-1\).

Step-by-Step Solution

Answer

The solutions are \( x_{1}(t) = 2e^{t} - 3te^{2t} \) and \( x_{2}(t) = -e^{2t} \).

Step 1: Identify the System of Differential Equations

The problem presents a system of linear differential equations: \( \frac{dx_{1}}{dt} = 1 \cdot x_{1} + 3 \cdot x_{2} \) and \( \frac{dx_{2}}{dt} = 2 \cdot x_{2} \). These equations can be rewritten as \( \frac{d\mathbf{x}}{dt} = A\mathbf{x} \), where \( A = \begin{bmatrix} 1 & 3 \ 0 & 2 \end{bmatrix} \) and \( \mathbf{x} = \begin{bmatrix} x_{1}(t) \ x_{2}(t) \end{bmatrix} \).

Step 2: Write the Matrix Solution Formula

For a linear system \( \frac{d\mathbf{x}}{dt} = A\mathbf{x} \), the solution is \( \mathbf{x}(t) = e^{At}\mathbf{x}(0) \), where \( \mathbf{x}(0) = \begin{bmatrix} x_{1}(0) \ x_{2}(0) \end{bmatrix} \). In this problem, \( x_{1}(0) = 2 \) and \( x_{2}(0) = -1 \), so \( \mathbf{x}(0) = \begin{bmatrix} 2 \ -1 \end{bmatrix} \).

Step 3: Compute the Exponential of Matrix A

To solve \( \mathbf{x}(t) = e^{At}\mathbf{x}(0) \), we find the matrix exponential \( e^{At} \). Since \( A \) is upper triangular, \( e^{At} = \begin{bmatrix} e^{t} & 3te^{2t} \ 0 & e^{2t} \end{bmatrix} \). This uses known results for matrix exponentials of upper triangular matrices.

Step 4: Calculate the Solution Using the Matrix Exponential

Multiply the matrix exponential by the initial condition vector: \( \mathbf{x}(t) = \begin{bmatrix} e^{t} & 3te^{2t} \ 0 & e^{2t} \end{bmatrix} \begin{bmatrix} 2 \ -1 \end{bmatrix} \). The result is \( \mathbf{x}(t) = \begin{bmatrix} 2e^{t} - 3te^{2t} \ -e^{2t} \end{bmatrix} \).

Step 5: Write Down the Individual Solutions

The solutions for each component are \( x_{1}(t) = 2e^{t} - 3te^{2t} \) and \( x_{2}(t) = -e^{2t} \). These are the expressions for \( x_{1}(t) \) and \( x_{2}(t) \) over time based on the given initial conditions.

Key Concepts

Initial-Value Problem

An initial-value problem focuses on finding a solution to a differential equation that satisfies specific given conditions at the outset, known as the initial conditions. In this context, we deal with the initial conditions of a system of linear differential equations. Differential equations are used to model how quantities dynamically change based on their current state.

In the given exercise, we have a system represented by \[ \frac{d \mathbf{x}}{dt} = A\mathbf{x} \]where the initial conditions are given as \[ x_{1}(0) = 2 \text{ and } x_{2}(0) = -1. \]This means that at time \( t = 0 \), the values of \( x_{1} \) and \( x_{2} \) are specified. Solving an initial-value problem involves finding \( x_{1}(t) \) and \( x_{2}(t) \) for all \( t \geq 0 \) such that they satisfy both the differential equations and the initial conditions.

• This ensures the solution not only fits the equations but also starts from the known situation given at \( t = 0 \).

Linear Algebra

Linear algebra provides the tools needed to analyze and solve systems of equations using concepts such as matrices and vectors, which are fundamental in representing and solving linear differential equations.

In our exercise, we can observe the matrix \( A \) as:\[ A = \begin{bmatrix} 1 & 3 \ 0 & 2 \end{bmatrix}\]This matrix shows how each component \( x_{1} \) and \( x_{2} \) interacts with themselves and with each other.
Matrix operations help in organizing the computations for the system:

• The matrix multiplication involved in computing \( \frac{d \mathbf{x}}{dt} \) is driven by these matrix entries.
• Understanding matrix types, like upper triangular matrices, simplifies complex operations, as seen in computing exponentials of matrices.

This structural insight allows us to efficiently solve differential equations by utilizing linear algebra techniques, making the solution process systematic and comprehensible.

Matrix Exponential

The matrix exponential \( e^{At} \) is a crucial concept in solving systems of linear differential equations, such as in our exercise. Understanding how to compute the matrix exponential allows for a direct solution expression of the differential equation system.

The exponential of a matrix \( A \) is defined similarly to the scalar exponential function but includes additional matrix operations:\[ e^{A} = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \ldots\]For the matrix \( A \) given:\[ A = \begin{bmatrix} 1 & 3 \ 0 & 2 \end{bmatrix},\]being upper triangular allows simplification to\[ e^{At} = \begin{bmatrix} e^{t} & 3te^{2t} \ 0 & e^{2t} \end{bmatrix}\]when using known theorems and properties for such matrices.

• This step is functionally vital as it transforms solving the differential equation from an analytical to an algebraic task, easily computed with matrix operations.
• The resultant matrix exponential simplifies obtaining \( \mathbf{x}(t) \) by straightforward multiplication with the initial condition vector.

Hence, mastering matrix exponentials significantly facilitates the resolution of initial-value problems involving differential equations.