Question

Suppose $$ \mathbf{y}_{1}=\left[\begin{array}{l} y_{11} \\ y_{21} \end{array}\right] \quad \text { and } \quad \mathbf{y}_{2}=\left[\begin{array}{l} y_{12} \\ y_{22} \end{array}\right] $$ are solutions of the homogeneous system $$ \mathbf{y}^{\prime}=A(t) \mathbf{y} $$ and define $$ Y=\left[\begin{array}{ll} y_{11} & y_{12} \\ y_{21} & y_{22} \end{array}\right] $$ (a) Show that \(Y^{\prime}=A Y\). (b) Show that if \(\mathbf{c}\) is a constant vector then \(\mathbf{y}=Y \mathbf{c}\) is a solution of \((\mathrm{A})\). (c) State generalizations of (a) and (b) for \(n \times n\) systems.

Short answer

Answer: When matrix Y is formed with two linearly independent solutions (y1, y2) of a homogeneous system, its derivative Y' is equal to AY, where A is a coefficient matrix of the system. In general, for an n × n system with n linearly independent solutions, Y' = AY.

Step-by-step solution

(a) Show that Y' = AY

To find the derivative of the matrix Y, we first need to define the matrix A(t) such that \(\boldsymbol{y}^{\prime}(t)=A(t) \boldsymbol{y}(t)\) for \(\boldsymbol{y}=[\boldsymbol{y_1}~\boldsymbol{y_2}]\). Then, we'll take the derivative of Y and compute the product AY to see if they are equal. Given that \(\boldsymbol{y_1}'=A(t)\boldsymbol{y_1}\) and \(\boldsymbol{y_2}'=A(t)\boldsymbol{y_2}\), we can compute Y': $$ Y' = \begin{bmatrix} y_{11}' & y_{12}' \\ y_{21}' & y_{22}' \end{bmatrix} = \begin{bmatrix} (A(t)y_{11}) & (A(t)y_{12}) \\ (A(t)y_{21}) & (A(t)y_{22}) \end{bmatrix} $$ Now we compute the matrix product AY: $$ A(t)Y = A(t) \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix} = \begin{bmatrix} (A(t)y_{11}) & (A(t)y_{12}) \\ (A(t)y_{21}) & (A(t)y_{22}) \end{bmatrix} $$ Since Y' and AY are equal, we have proved that \(Y'= AY\).

(b) Show that y = Yc is a solution of the homogeneous system

To show that \(\boldsymbol{y} = Y\boldsymbol{c}\) is a solution of the homogeneous system, we will first compute the derivative \(\boldsymbol{y}'\) and check if it is equal to \(A(t) \boldsymbol{y}(t)\). Given that $\boldsymbol{y} = Y\boldsymbol{c} = \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}\(, we can compute the derivative of \)\boldsymbol{y}$: $$ \boldsymbol{y}' = \begin{bmatrix} y_{11}' & y_{12}' \\ y_{21}' & y_{22}' \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} $$ Since \(Y'=AY\) and \(Y'=\begin{bmatrix}y_{11}' & y_{12}' \\ y_{21}' & y_{22}'\end{bmatrix}\), we can rewrite \(\boldsymbol{y}'\) as: $$ \boldsymbol{y}' = \begin{bmatrix} (A(t)y_{11}) & (A(t)y_{12}) \\ (A(t)y_{21}) & (A(t)y_{22}) \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}=A \begin{bmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} = A \boldsymbol{y} $$ As we can see, \(\boldsymbol{y}' = A \boldsymbol{y}\), which means that \(\boldsymbol{y} = Y \boldsymbol{c}\) is indeed a solution of the homogeneous system.

(c) State generalizations of (a) and (b) for \(n \times n\) systems

The generalizations of the above properties for an \(n \times n\) system would be: (a) If we have n linearly independent solutions (\(\boldsymbol{y_1}, \boldsymbol{y_2}, \ldots, \boldsymbol{y_n}\)) to a homogeneous system such that \(\boldsymbol{y_i}' = A(t)\boldsymbol{y_i}\), we can form an n × n matrix Y using these solutions as columns. Then \(Y' = AY\). (b) If \(\boldsymbol{c}\) is an n-dimensional constant vector, then a linear combination \(\boldsymbol{y} = Y\boldsymbol{c}\) of these n solutions is also a solution of the homogeneous system.

Key concepts

Matrix Solutions

In the context of homogeneous systems of differential equations, matrix solutions are an elegant way of expressing the relationships between systems of equations. Each row or column of a matrix represents a solution to the differential equation. This is particularly useful because it allows for a compact, organized means of representing multiple solutions simultaneously. For example, if we have two solutions to a homogeneous system, these solutions can be organized into a matrix to form what is known in mathematics as a matrix solution.

The matrix that is formed by these solutions allows us to see how transformations, like taking derivatives, can be applied to each solution simultaneously. In the original exercise, \( Y \) represents a matrix composed of vector solutions \( \mathbf{y}_1 \) and \( \mathbf{y}_2 \). By proving that the derivative of this matrix \( Y' = A(t)Y \), we demonstrate an important property related to how systems of equations interact with differential operators. This property forms the foundation for solving larger \( n \times n \) systems through matrix factorization and similar methodologies.

Vector Equations

Vector equations are an integral part of understanding systems of differential equations. A vector equation represents a solution in the form of vectors instead of scalars, which is highly beneficial when dealing with multiple interrelated differential equations.

When handling a homogeneous system of differential equations, the entire set of solutions is often expressed via vectors. Each vector holds the values of the system’s variables as they change over time, making complex relationships easier to manage. For example, given vectors \( \mathbf{y}_1 \) and \( \mathbf{y}_2 \), we see that they can be utilized to form a solution in the matrix \( Y \). The subsequent vector equation becomes useful, allowing us to compute derivatives and transformations for all parts of the solution simultaneously.

Using vector equations not only simplifies calculations but also clarifies the structure of complex systems, enabling mathematicians and scientists to draw broader conclusions from their studies of dynamic systems.

General Solutions

The notion of general solutions in the realm of differential equations is crucial because it encompasses the sum of all possible particular solutions to a system. In the study of homogeneous systems, when we derive a general solution, we demonstrate that it isn't just one solution but a family of solutions dependent on a set of parameters.

This is reflected in the concept of creating a general solution using arbitrary constants that can be adjusted depending on initial conditions or specific requirements. In the provided textbook solution, \( \mathbf{y} = Y \mathbf{c} \) where \( \mathbf{c} \) is a constant vector, gives us a generalized form. This form accommodates every possible solution via linear combination, wherein \( \mathbf{c} \) defines how these solutions are weighted.

General solutions provide a comprehensive look at all potential shapes the solution might take, granting flexibility in applying these mathematical models to real-world scenarios where initial conditions and constraints might vary. Understanding general solutions helps in predicting system behavior under various circumstances and serves as a fundamental tool in mathematical modeling and simulations.