Question

Rewrite the system in matrix form and verify that the given vector function satisfies the system for any choice of the constants \(c_{1}\) and \(c_{2}\). (c) \(\begin{array}{l}y_{1}^{\prime}=-4 y_{1}-10 y_{2} \\ y_{2}^{\prime}=3 y_{1}+7 y_{2} ;\end{array} \quad \mathbf{y}=c_{1}\left[\begin{array}{l}-5 \\\ 3\end{array}\right] e^{2 t}+c_{2}\left[\begin{array}{l}2 \\\ -1\end{array}\right] e^{t}\) (d) \(\begin{aligned} y_{1}^{\prime} &=2 y_{1}+y_{2} \\ y_{2}^{\prime} &=y_{1}+2 y_{2} ; \end{aligned} \quad \mathbf{y}=c_{1}\left[\begin{array}{l}1 \\\ 1\end{array}\right] e^{3 t}+c_{2}\left[\begin{array}{l}1 \\\ -1\end{array}\right] e^{t}\)

Short answer

#Answer# Both given vector functions satisfy their respective systems of differential equations for any values of constants \(c_{1}\) and \(c_{2}\). In the case of (c), the vector function \(\textbf{y}(t)=c_1\left[\begin{array}{l}-5 \\\ 3\end{array}\right] e^{2 t} + c_2\left[\begin{array}{l}2 \\\ -1\end{array}\right] e^{t}\) satisfies the system of differential equations \(y_1' = -4y_1 - 10y_2\) and \(y_2' = 3y_1 + 7y_2\). In the case of (d), the vector function \(\textbf{y}(t)=c_{1}\left[\begin{array}{l}1 \\\ 1\end{array}\right]e^{3t} + c_{2}\left[\begin{array}{l}1 \\\ -1\end{array}\right]e^t\) satisfies the system of differential equations \(y_1' = 2y_1 + y_2\) and \(y_2' = y_1 + 2y_2\).

Step-by-step solution

Rewrite in matrix form

Rewrite the given system of differential equations into matrix form: \(\mathbf{y}^{\prime}=A\mathbf{y}\) \(\begin{bmatrix}y_{1}^{\prime}\\y_{2}^{\prime}\end{bmatrix}=\begin{bmatrix}-4 & -10 \\ 3 & 7\end{bmatrix}\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}\)

Plug in the given vector function for y and its derivative

Find the derivative of the given vector function: \(\mathbf{y}^{\prime}=c_{1}\left[\begin{array}{l}-5 \\\ 3\end{array}\right] (2 e^{2 t})+c_{2}\left[\begin{array}{l}2 \\\ -1\end{array}\right] (e^{t})\) Plug the given vector function \(\mathbf{y}\) and its derivative \(\mathbf{y}^{\prime}\) into the matrix equation: \(\begin{bmatrix}c_1(-5e^{2t}+2e^t)c_2 \\ c_1(6e^{2t}-e^t)c_2\end{bmatrix} = \begin{bmatrix}-4 & -10 \\ 3 & 7\end{bmatrix} \begin{bmatrix}c_{1}(-5e^{2t}+2e^{t}c_2) \\ c_{1}(3e^{2t}-e^{t}c_2)\end{bmatrix}\)

Show that the given matrix equation holds true

Verify that the equations are the same for any values of \(c_{1}\) and \(c_{2}\) by simplifying the matrix multiplication: \(\begin{bmatrix}-4(-5c_1e^{2t}+2c_2e^t)-10(3c_1e^{2t}-c_2e^t) \\ 3(-5c_1e^{2t}+2c_2e^t)+7(3c_1e^{2t}-c_2e^t)\end{bmatrix}=\begin{bmatrix}c_1(-5e^{2t}+2e^t)c_2 \\ c_1(6e^{2t}-e^t)c_2\end{bmatrix}\) The matrix equation is true for any values of \(c_{1}\) and \(c_{2}\), which means that the given vector function satisfies the given system of differential equations. Now, consider case (d).

Rewrite in matrix form

Rewrite the given system of differential equations into matrix form: \(\mathbf{y}^{\prime}=A\mathbf{y}\) \(\begin{bmatrix}y_{1}^{\prime}\\y_{2}^{\prime}\end{bmatrix}=\begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix}\begin{bmatrix}y_{1}\\y_{2}\end{bmatrix}\)

Plug in the given vector function for y and its derivative

Find the derivative of the given vector function: \(\mathbf{y}^{\prime}=c_{1}\left[\begin{array}{l}1 \\\ 1\end{array}\right] (3 e^{3 t})+c_{2}\left[\begin{array}{l}1 \\\ -1\end{array}\right] (e^{t})\) Plug the given vector function \(\mathbf{y}\) and its derivative \(\mathbf{y}^{\prime}\) into the matrix equation: \(\begin{bmatrix}c_1(3 e^{3 t})+c_2(e^{t}) \\ c_1(3 e^{3 t})-c_2(e^t)\end{bmatrix} = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix} \begin{bmatrix}c_1(e^{3 t})+c_2(e^{t}) \\ c_1(e^{3 t})-c_2(e^{t})\end{bmatrix}\)

Show that the given matrix equation holds true

Verify that the equations are the same for any values of \(c_{1}\) and \(c_{2}\) by simplifying the matrix multiplication: \(\begin{bmatrix}2(c_1e^{3t}+c_2e^t)+1(c_1e^{3t}-c_2e^t) \\ 1(c_1e^{3t}+c_2e^t)+2(c_1e^{3t}-c_2e^t)\end{bmatrix}=\begin{bmatrix}c_1(3e^{3t})+c_2(e^t) \\ c_1(3e^{3t})-c_2(e^t)\end{bmatrix}\) The matrix equation is true for any values of \(c_{1}\) and \(c_{2}\), which means that the given vector function satisfies the given system of differential equations.

Key concepts

Systems of Differential Equations

Systems of differential equations are collections of two or more equations involving derivatives of one or more dependent variables with respect to one or more independent variables. When dealing with multiple variables and their rates of change, these systems allow us to describe complex dynamical phenomena.

For many physical systems, individual components are interdependent, and their state variables evolve according to these systems of equations. The functionality of an electric circuit with multiple components, the kinetics of chemical reactions, and the behavior of predator-prey populations in biology are examples of where systems of differential equations play a crucial role. By solving these systems, we can predict and understand the evolution of these systems over time.

Matrix Algebra

Matrix algebra provides a compact and efficient way to represent and manipulate systems of linear equations. A matrix is essentially a rectangular array of numbers, called elements, arranged in rows and columns, which acts as a single object.

When writing systems of differential equations in matrix form, the coefficients of the variables are organized into a coefficient matrix, while the variables and their derivatives are represented as vector matrices. This simplifies operations such as adding and multiplying systems of equations and is especially powerful when combined with concepts like eigenvalues and eigenvectors, which we will explore shortly.

Eigenvalues and Eigenvectors

Within the realm of linear algebra, eigenvalues and eigenvectors are instrumental in understanding the behavior of linear transformations represented by matrices. An eigenvector of a matrix is a non-zero vector that only changes by a scalar factor, called an eigenvalue, when that linear transformation is applied.

In the context of systems of differential equations, eigenvalues can determine the system's stability and the type of growth, decay, or oscillations of the solutions. Eigenvectors, on the other hand, are related to the directions of these behaviors in the space of variables. Thus, finding the eigenvalues and eigenvectors of the coefficient matrix can often give us qualitative information about the solutions, even when finding an explicit solution is challenging.

Exponential Functions

Exponential functions, of the form f(t) = e^{kt} where e is the base of the natural logarithm and k is a constant, are ubiquitous in the study of differential equations. Their importance lies in their property of being their derivatives, up to a constant factor.

These functions describe processes that change at a rate proportional to their current value—such as radioactive decay, population growth, and compound interest. In solving linear systems of differential equations, exponential functions frequently emerge as part of the general solution, with the exponentials often being raised to the power of eigenvalues multiplied by the independent variable.