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 \(\mathbf{g}(t)\) $$ \begin{aligned} &y_{1}^{\prime}=t^{-1} y_{1}+\left(t^{2}+1\right) y_{2}+t \\ &y_{2}^{\prime}=4 y_{1}+t^{-1} y_{2}+8 t \ln t \end{aligned} $$

Short answer

Question: Determine the matrices \(P(t)\) and \(\mathbf{g}(t)\) that express the given system of linear differential equations in the form \(\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t)\) for the following system: $$ \begin{cases} y_{1}^{\prime} = \frac{1}{t}y_{1} + (t^{2}+1)y_{2} + t\\ y_{2}^{\prime} = 4y_{1} + \frac{1}{t}y_{2} + 8t \ln t \end{cases} $$ Answer: \(P(t) = \begin{pmatrix} t^{-1} & t^{2}+1 \\ 4 & t^{-1} \end{pmatrix}\) and \(\mathbf{g}(t) = \begin{pmatrix} t \\ 8t \ln t \end{pmatrix}\).

Step-by-step solution

Rewrite the system as a matrix equation

We can rewrite the system of linear differential equations as a matrix equation of the form \(\mathbf{y}^{\prime}=A(t) \mathbf{y}+\mathbf{b}(t)\), where \(\mathbf{y} = \begin{pmatrix} y_1 \\ y_2\end{pmatrix}\), \(A(t)\) is a \(2 \times 2\) matrix, and \(\mathbf{b}(t)\) is a \(2\times 1\) vector. For the given system, we have: $$ \begin{pmatrix} y_{1}^{\prime} \\ y_{2}^{\prime} \end{pmatrix} = \begin{pmatrix} t^{-1} & t^{2}+1 \\ 4 & t^{-1} \end{pmatrix} \begin{pmatrix} y_{1} \\ y_{2} \end{pmatrix} + \begin{pmatrix} t \\ 8t \ln t \end{pmatrix} $$

Identify the P(t) matrix and g(t) vector

Comparing the matrix equation above with the general form \(\mathbf{y}^{\prime}=P(t) \mathbf{y}+\mathbf{g}(t)\), we can identify the \(P(t)\) matrix and \(\mathbf{g}(t)\) vector as follows: $$ P(t) = \begin{pmatrix} t^{-1} & t^{2}+1 \\ 4 & t^{-1} \end{pmatrix} $$ and $$ \mathbf{g}(t) = \begin{pmatrix} t \\ 8t \ln t \end{pmatrix} $$ Therefore, the matrices \(P(t)\) and \(\mathbf{g}(t)\) that express the given system of linear differential equations in the required form are $$ P(t) = \begin{pmatrix} t^{-1} & t^{2}+1 \\ 4 & t^{-1} \end{pmatrix} \quad\text{and}\quad \mathbf{g}(t) = \begin{pmatrix} t \\ 8t \ln t \end{pmatrix}. $$

Key concepts

Systems of Differential Equations

Understanding how to approach and solve systems of differential equations is crucial for students tackling advanced math topics. These systems consist of multiple equations that describe a set of related functions and their derivatives.

For instance, considering two functions, one might find equations that express their rates of change—often in the context of time—with those rates possibly being influenced by both functions. In the example provided, the system of equations involves two functions, \(y_1(t)\) and \(y_2(t)\), where each function's derivative is given in terms of both \(y_1\) and \(y_2\) and the independent variable \(t\).

When approaching such problems, a systemic strategy involves organizing the information in a way that leverages matrix algebra, transforming seemingly complex relationships into a manageable form. This strategy paves the way for matrix representation, which we'll discuss next.

Matrix Representation of Differential Equations

The matrices in these representations capture the coefficients of the functions within the system, providing a compact way to express the entire system. For our specific problem, \(A(t)\) becomes \(P(t)\), and \(\mathbf{b}(t)\) becomes \(\mathbf{g}(t)\), where \(P(t)\) neatly encapsulates the coefficients of \(y_1\) and \(y_2\), while \(\mathbf{g}(t)\) represents the nonhomogeneous part or the 'forcing' of the system. The power of this approach lies in enabling us to utilize matrix operations, such as finding eigenvalues and eigenvectors, which can greatly facilitate the solution of differential systems.

Homogeneous and Nonhomogeneous Differential Equations

Dividing differential equations into homogeneous and nonhomogeneous types aids in their analysis and solution. A homogeneous differential equation is one where every term is a function of the dependent variable (such as \(y\)) and its derivatives. In contrast, a nonhomogeneous differential equation includes terms that are not functions of the dependent variable or its derivatives—these are often called forcing functions.

In our example, \(\mathbf{g}(t)\) introduces a nonhomogeneous component because it includes terms that are not merely coefficients of \(y_1\) and \(y_2\). As such, the term \(t\) and \(8t \ln t\) represent external influences or inputs to the system that differentiate it from a purely homogeneous equation, where \(\mathbf{g}(t)\) would be zero. Learning to distinguish between these types of equations is foundational for effectively applying different methods for finding solutions—the homogeneous part often relates to the complementary solution, while the nonhomogeneous part requires a particular solution.

Method of Undetermined Coefficients

The method of undetermined coefficients is a practical technique for finding particular solutions to nonhomogeneous linear differential equations. This method assumes a form for the particular solution, then determines the unknown coefficients by plugging the assumed solution into the differential equation and equating coefficients.

The key is selecting the correct form for the particular solution, which depends on the nonhomogeneous terms present in \(\mathbf{g}(t)\). For example, if \(\mathbf{g}(t)\) contains polynomials, exponentials, sines, or cosines, the assumed particular solution should mirror these forms. After determining the form, calculations will reveal the specific coefficients that make the assumed solution valid.

The method of undetermined coefficients is a formulaic approach, but it requires a careful consideration of the nonhomogeneous terms and sometimes a bit of ingenuity, especially in the case of systems of differential equations. Through practice and a strong grasp of algebra, students can master this indispensable analytic tool.