Question

Each of the systems of linear differential equations can be expressed in the form \(\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t) .\) Determine \(P(t)\) and \(\mathbf{g}(t)\) $$ A^{\prime}(t)=\left[\begin{array}{cc} 2 t & 1 \\ \cos t & 3 t^{2} \end{array}\right], \quad A(0)=\left[\begin{array}{rr} 2 & 5 \\ 1 & -2 \end{array}\right] $$

Short answer

Based on the given matrix \(A'(t)\) and initial condition \(A(0)\), the system of linear differential equations can be expressed in the form \(\mathbf{y}' = P(t) \mathbf{y} + \mathbf{g}(t)\) with \(P(t) = \begin{bmatrix} 2t & 1 \\ \cos{t} & 3t^2 \end{bmatrix}\) and \(\mathbf{g}(t) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\).

Step-by-step solution

Write down the given information

We have \(A'(t) = \begin{bmatrix} 2t & 1 \\ \cos{t} & 3t^2 \end{bmatrix}\) and \(A(0) = \begin{bmatrix} 2 & 5 \\ 1 & -2 \end{bmatrix}\).

Write down the general form of the given system

The general form of the given system is \(\mathbf{y}' = P(t) \mathbf{y} + \mathbf{g}(t)\). To find \(P(t)\) and \(\mathbf{g}(t)\), we will compare this form with the form of the matrix A.

Determine the form of matrix P(t) and vector g(t) using the given matrix A'(t)

Since the columns of A'(t) must correspond to the elements of P(t) multiplied by the coordinate vector \(\mathbf{y}\), we can write the system as \(\frac{d\mathbf{y}}{dt} = \begin{bmatrix} p_{11}(t) & p_{12}(t) \\ p_{21}(t) & p_{22}(t) \end{bmatrix} \mathbf{y} + \begin{bmatrix} g_1(t) \\ g_2(t) \end{bmatrix}\). Now, by comparing the given matrix \(A'(t)\) with the form \(\begin{bmatrix} p_{11}(t) & p_{12}(t) \\ p_{21}(t) & p_{22}(t) \end{bmatrix} \mathbf{y} + \begin{bmatrix} g_1(t) \\ g_2(t) \end{bmatrix}\), we get the following equations: 1. \(2t = p_{11}(t)y_1 + p_{12}(t)y_2 + g_1(t)\) 2. \(1 = p_{21}(t)y_1 + p_{22}(t)y_2 + g_2(t)\) 3. \(\cos{t} = p_{11}(t)y_1' + p_{12}(t)y_2'\) 4. \(3t^2 = p_{21}(t)y_1' + p_{22}(t)y_2'\)

Solve for the entries of P(t) and g(t)

Since the system is homogeneous, we can determine that \(\mathbf{g}(t) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\). Now we can set up the following equations for the entries of \(P(t)\): 1. \(p_{11}(t) = 2t\) 2. \(p_{12}(t) = 1\) 3. \(p_{21}(t) = \cos{t}\) 4. \(p_{22}(t) = 3t^2\) Thus, \(P(t) = \begin{bmatrix} 2t & 1 \\ \cos{t} & 3t^2 \end{bmatrix}\) and \(\mathbf{g}(t) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\). The system of linear differential equations can be expressed in the form \(\mathbf{y}' = P(t) \mathbf{y} + \mathbf{g}(t)\) with \(P(t) = \begin{bmatrix} 2t & 1 \\ \cos{t} & 3t^2 \end{bmatrix}\) and \(\mathbf{g}(t) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}\).

Key concepts

Matrix Representation

When dealing with systems of linear differential equations, transforming them into a matrix framework is highly beneficial. This involves expressing the system in the form \( \mathbf{y}' = P(t) \mathbf{y} + \mathbf{g}(t) \), where \( P(t) \) is a matrix that encompasses the coefficients of the system and \( \mathbf{g}(t) \) is a vector that represents any non-homogeneous part.

• In our exercise, the matrix \( A'(t) \) is given, which represents the derivative of the state matrix over time. It is structured as \( \begin{bmatrix} 2t & 1 \ \cos{t} & 3t^2 \end{bmatrix} \).
• We aim to identify \( P(t) \), which corresponds directly to the given derivative matrix, because the system is homogeneous with \( \mathbf{g}(t) = \begin{bmatrix} 0 \ 0 \end{bmatrix} \).
• Consequently, the matrix \( P(t) = \begin{bmatrix} 2t & 1 \ \cos{t} & 3t^2 \end{bmatrix} \).

Homogeneous Systems

A system of linear differential equations can be either homogeneous or non-homogeneous. In a homogeneous system, there is no external force or addition, meaning \( \mathbf{g}(t) \) in the equation \( \mathbf{y}' = P(t) \mathbf{y} + \mathbf{g}(t) \) is a zero vector.

• This implies that all changes in the system are due to the system's inherent properties, represented by matrix \( P(t) \).
• For the problem at hand, the solution reveals that \( \mathbf{g}(t) = \begin{bmatrix} 0 \ 0 \end{bmatrix} \), confirming the homogeneity of the system.
• Homogeneous systems often simplify the complexity involved, as they focus solely on internal dynamics without external stimuli.

Understanding whether a system is homogeneous helps determine the approach and tools we use for solving it.

Differential Equation Solutions

Solving a system of linear differential equations involves finding a function or set of functions that satisfy the system's equation.

• The core idea here is to express derivatives concretely in terms of equations that can then be solved.
• In homogeneous systems like ours, where \( \mathbf{g}(t) = \begin{bmatrix} 0 \ 0 \end{bmatrix} \), solutions typically involve determining eigenvalues and eigenvectors of the matrix \( P(t) \).
• The eigenvalues provide insight into the stability and behavior of solutions over time, offering a pathway to predict system dynamics.

The solution usually involves multiple steps, integrating knowledge of linear algebra and calculus to piece together the function \( \mathbf{y}(t) \) that successfully represents the state of the system through time, maintaining alignment with \( P(t) \).