Question

Rewrite the given boundary value problem as an equivalent boundary value problem for a first order system. Your rewritten boundary value problem should have the form of equation (1): $$ \begin{aligned} &\mathbf{y}^{\prime}=A(t) \mathbf{y}+\mathbf{g}(t), \quad a

Short answer

Question: Rewrite the given boundary value problem involving a second-order linear differential equation into an equivalent first-order system. Given: $$ y''(t) - 2y'(t) + y(t) = \cos(2t) $$ Boundary conditions: $$ 2y(0) - y'(0) = -1, \quad y(1) + y'(1) = 2 $$ Solution: The equivalent first-order system is: $$ \begin{aligned} &\mathbf{y}'(t) = A(t) \mathbf{y}(t) + \mathbf{g}(t), \quad 0<t<1 \\ &P^{[a]} \mathbf{y}(0) + P^{[b]} \mathbf{y}(1) = \alpha \end{aligned} $$ with $$ A(t) = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}, \quad \mathbf{g}(t) = \begin{pmatrix} 0 \\ \cos(2t) \end{pmatrix}, \quad P^{[a]} = \begin{pmatrix} 2 & -1 \\ 0 & 0 \end{pmatrix}, $$ $$ P^{[b]} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}, \quad \text{and} \quad \alpha = \begin{pmatrix} -1 \\ 2 \end{pmatrix} $$

Step-by-step solution

Convert the second-order equation into a system of first-order equations

Let's define two new functions: $$ \mathbf{y}(t) = \begin{pmatrix} y_1(t) \\ y_2(t) \end{pmatrix} = \begin{pmatrix} y(t) \\ y'(t) \end{pmatrix} $$ With this new notation, the second-order differential equation can be represented as a system of first-order equations: $$ \begin{aligned} y_1'(t) &= y_2(t) \\ y_2'(t) &= 2y_2(t) - y_1(t) + \cos(2t) \end{aligned} $$ Now we can write this system in matrix form: $$ \mathbf{y}'(t) = A(t) \mathbf{y}(t) + \mathbf{g}(t), $$ Where $$ A(t) = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix} \quad \text{and} \quad \mathbf{g}(t) = \begin{pmatrix} 0 \\ \cos(2t) \end{pmatrix} $$

Rewrite boundary conditions in matrix form

We are given the following boundary conditions: $$ 2 y(0) - y'(0) = -1, \quad y(1) + y'(1) = 2 $$ Using the definitions of \(y_1(t)\) and \(y_2(t)\), we can rewrite these boundary conditions as: $$ 2 y_1(0) - y_2(0) = -1, \quad y_1(1) + y_2(1) = 2 $$ Now we can write the boundary conditions in matrix form: $$ P^{[a]}\mathbf{y}(0) + P^{[b]}\mathbf{y}(1) = \alpha $$ Where $$ P^{[a]} = \begin{pmatrix} 2 & -1 \\ 0 & 0 \end{pmatrix}, \quad P^{[b]} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}, \quad \text{and} \quad \alpha = \begin{pmatrix} -1 \\ 2 \end{pmatrix} $$ Now we have rewritten the boundary value problem as an equivalent first-order system: $$ \begin{aligned} &\mathbf{y}'(t) = A(t) \mathbf{y}(t) + \mathbf{g}(t), \quad 0<t<1 \\ &P^{[a]} \mathbf{y}(0) + P^{[b]} \mathbf{y}(1) = \alpha \end{aligned} $$ with $$ A(t) = \begin{pmatrix} 0 & 1 \\ -1 & 2 \end{pmatrix}, \quad \mathbf{g}(t) = \begin{pmatrix} 0 \\ \cos(2t) \end{pmatrix}, \quad P^{[a]} = \begin{pmatrix} 2 & -1 \\ 0 & 0 \end{pmatrix}, $$ $$ P^{[b]} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}, \quad \text{and} \quad \alpha = \begin{pmatrix} -1 \\ 2 \end{pmatrix} $$

Key concepts

First-order Systems

First-order systems of differential equations are crucial in understanding complex dynamic systems. By breaking down higher-order equations into a series of first-order equations, we can analyze the system from a new perspective. In our exercise, we started by introducing a set of functions to define a first-order system. This included:

• Defining the vector \(\mathbf{y}(t)\), which contains both \(y(t)\) and its derivative \(y'(t)\).
• Establishing a relationship where \(y_1'(t) = y_2(t)\) and \(y_2'(t) = 2y_2(t) - y_1(t) + \cos(2t)\).

The primary advantage of converting to a first-order system is the uniformity it provides, simplifying analysis and solution finding. Once in first-order form, numerical and analytical techniques for solving these systems become more accessible, allowing for easier computation and an increased understanding of the behaviors illustrated in the system.

Second-order Differential Equations

Second-order differential equations are pervasive in physical sciences and engineering. These equations often represent systems with motion or other dynamic properties involving acceleration, as the order corresponds to the highest derivative, which is the second derivative in this case. The initial equation given:\[y'' - 2y' + y = \cos(2t)\]represents a boundary value problem, typical of a system experiencing forces or changes in velocity over time.
When dealing with such equations, the method of converting into a system of first-order equations, as shown in our exercise, simplifies both the solving process and the theoretical understanding of the system's dynamics. In practice, this means turning the initial differential problem into one that can be more easily computed, especially with computational software. By re-expressing it into a first-order system, we can handle complex interdependencies within the model more effectively.

Matrix Representation

Matrix representation plays a pivotal role in modern mathematics and engineering applications. In the context of this exercise, we focused on presenting our system of first-order differential equations in matrix form. This involved defining specific matrices, such as:

• The matrix \(A(t)\), which encapsulates the relationships between derivative terms.
• A vector \(\mathbf{g}(t)\) that accounts for the external force or function \(\cos(2t)\).

By transforming these differential equations into a matrix form, we open up the application of linear algebra techniques. This representation allows us to leverage tools such as eigenvectors and eigenvalues, making it easier to analyze the stability and behavior of the system. Moreover, this matrix approach facilitates solving the system using computational resources, which can handle large matrices efficiently.

Differential Equation Conversion

Differential equation conversion is the process of transforming one type of differential equation into another form, often done to simplify the solving process or to enable the use of specific analytical techniques. In our exercise, the conversion of a second-order differential equation into a system of first-order equations was demonstrated. This involved several steps:

• Defining new functions to separate the original equation into distinct first-order equations.
• Utilizing matrices to elegantly express these new relationships as a cohesive system.

This conversion is crucial because it transforms a complex problem into a series of simpler problems, manageable with conventional linear algebra methods. When solving boundary value problems, this systematic conversion is often a preliminary step in finding numerical solutions and provides a clearer visual structure to understand the dynamics at play. This approach is foundation to many fields, simplifying the application of various mathematical techniques that provide solutions to otherwise insurmountable problems.