Question

Exercises 11-17 dealt with rewriting a single scalar equation as a first order system. Frequently, however, we need to convert systems of higher order equations into a single first order system. In each exercise, rewrite the given system of two second order equations as a system of four first order linear equations of the form \(\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t)\). In each exercise, use the following change of variables and identify \(P(t)\) and \(\mathbf{g}(t)\) : $$ \mathbf{y}(t)=\left[\begin{array}{l} y_{1}(t) \\ y_{2}(t) \\ y_{3}(t) \\ y_{4}(t) \end{array}\right]=\left[\begin{array}{l} y(t) \\ y^{\prime}(t) \\ z(t) \\ z^{\prime}(t) \end{array}\right] $$ $$ \begin{aligned} &y^{\prime \prime}=7 y^{\prime}+4 y-8 z+6 z^{\prime}+t^{2} \\ &z^{\prime \prime}=5 z^{\prime}+2 z-6 y^{\prime}+3 y-\sin t \end{aligned} $$

Short answer

Question: Rewrite the following system of second-order linear differential equations as a single first-order linear system of equations using the change of variables. $$ \begin{aligned} &y^{\prime \prime}=7 y^{\prime}+4 y-8 z+6 z^{\prime}+t^{2} \\\ &z^{\prime \prime}=5 z^{\prime}+2 z-6 y^{\prime}+3 y-\sin t \end{aligned} $$ Answer: Using the change of variables, the given system of second-order equations can be rewritten as a single first-order linear system of equations: $$\mathbf{y}^{\prime} = P(t) \mathbf{y} + \mathbf{g}(t)$$ where $$ P(t) = \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\\ 4 & 7 & -8 & 6 \\\ 0 & 0 & 0 & 1 \\\ 3 & -6 & 2 & 5 \end{array}\right], \mathbf{g}(t) = \left[\begin{array}{c} 0 \\\ t^{2} \\\ 0 \\\ -\sin t \end{array}\right] $$

Step-by-step solution

Change of Variables

In the given vector form \(\mathbf{y}(t)\), we have the following relationships: $$y_1(t) = y(t)$$ $$y_2(t) = y'(t)$$ $$y_3(t) = z(t)$$ $$y_4(t) = z'(t)$$ Now we will substitute the change of variables into the original equations and write the first-order relationships:

First-Order Relationships

We start by writing the first order relationships from the given vector form: $$y^{\prime} = y_2(t)$$ $$y^{\prime\prime} = y_2^{\prime}(t)$$ $$z^{\prime} = y_4(t)$$ $$z^{\prime\prime} = y_4^{\prime}(t)$$ Now, we have: $$y_2^{\prime} = 7 y_2 + 4 y_1 - 8 y_3 + 6 y_4 + t^2$$ $$y_4^{\prime} = 5 y_4 + 2 y_3 - 6 y_2 + 3 y_1 - \sin t$$

Rewrite the System as a First-Order System

Using the first-order relationships, we rewrite the system in the form \(\mathbf{y}^{\prime} = P(t) \mathbf{y} + \mathbf{g}(t)\) as: $$ \mathbf{y}^{\prime} = \left[\begin{array}{l} y_{1}^{\prime}(t) \\\ y_{2}^{\prime}(t) \\\ y_{3}^{\prime}(t) \\\ y_{4}^{\prime}(t) \end{array}\right] = \left[\begin{array}{l} y_2 \\\ 7 y_2 + 4 y_1 - 8 y_3 + 6 y_4 + t^{2} \\\ y_4 \\\ 5 y_4 + 2 y_3 - 6 y_2 + 3 y_1 - \sin t \end{array}\right] $$ Now, we separate \(P(t)\) and \(\mathbf{g}(t)\):

Identify \(P(t)\) and \(\mathbf{g}(t)\)

We identify \(P(t)\) and \(\mathbf{g}(t)\) as: $$ P(t) = \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\\ 4 & 7 & -8 & 6 \\\ 0 & 0 & 0 & 1 \\\ 3 & -6 & 2 & 5 \end{array}\right], \mathbf{g}(t) = \left[\begin{array}{c} 0 \\\ t^{2} \\\ 0 \\\ -\sin t \end{array}\right] $$ Thus, we have successfully rewritten the given system of two second-order equations as a single first-order linear system of equations: $$\mathbf{y}^{\prime} = P(t) \mathbf{y} + \mathbf{g}(t)$$

Key concepts

Change of Variables

When dealing with complex systems of differential equations, one effective strategy is to apply a change of variables. This technique simplifies the equations, making them easier to handle. In this exercise, you are given two second-order differential equations that describe some system. To make these equations simpler and consistent, a transformation is applied by representing the functions and their derivatives in a new form.

The new variables are introduced as follows:

• \( y_1(t) = y(t) \)
• \( y_2(t) = y'(t) \)
• \( y_3(t) = z(t) \)
• \( y_4(t) = z'(t) \)

Applying these transformations allows us to express the second-order equations in terms of first-order derivatives, making the system easier to solve and analyze. This step is crucial as it allows the rewriting of the system to exploit mathematical techniques designed primarily for first-order systems.

Second-order Equations

A second-order equation is a type of differential equation where the highest derivative is the second derivative. For the given problem, you are faced with two such equations:

• \( y^{\prime \prime} = 7y^{\prime} + 4y - 8z + 6z^{\prime} + t^2 \)
• \( z^{\prime \prime} = 5z^{\prime} + 2z - 6y^{\prime} + 3y - \sin t \)

These complex relationships describe how the functions \(y(t)\) and \(z(t)\) interact not only with their immediate forms but also with their rates of change. Such equations are prevalent in physical systems where acceleration or relative changes are considered. The task is to reduce these second-order equations to first-order equations, making them more manageable to solve using the new variables proposed in the change of variables.

Matrix Representation

The key to simplifying and solving systems of equations is often the use of a matrix representation. In this context, once the original complex system is rewritten as a first-order system, it can be represented using matrices.

The system is expressed in a compact form such as \( \mathbf{y}^{\prime} = P(t)\mathbf{y} + \mathbf{g}(t) \). This helps in organizing the equations by grouping all coefficients into a matrix. In our solution:- The matrix \(P(t)\) is as follows:\[ P(t) = \begin{bmatrix} 0 & 1 & 0 & 0 \ 4 & 7 & -8 & 6 \ 0 & 0 & 0 & 1 \ 3 & -6 & 2 & 5 \end{bmatrix} \]- The vector \(\mathbf{g}(t)\) is:\[ \mathbf{g}(t) = \begin{bmatrix} 0 \ t^2 \ 0 \ -\sin t \end{bmatrix} \]
Matrices efficiently summarize the relationships between variables and equations, allowing us easier ways to manipulate and solve them using known techniques in linear algebra.

Linear Systems

Once rewritten, the equations fit into the category of a linear system of first-order differential equations. Linear systems are advantageous because they adhere to superposition principles. This makes them generally easier to analyze and solve than their nonlinear counterparts. In this setting, the system of equations can be expressed as:

• \( \mathbf{y}^{\prime} = P(t) \mathbf{y} + \mathbf{g}(t) \)

This linear form is beneficial as many theoretical results and numerical methods are available for solving such systems. They allow us to explore solutions by examining eigenvalues, eigenvectors, or employing numerical solvers to approximate them efficiently.

This exercise emphasizes converting higher-order systems into linear ones, demonstrating their versatility and usefulness in modeling real-world scenarios.