Question

Use the operator method described in this section to find the general solution of each of the linear systems in Exercises 1-26. $$ \begin{aligned} &x^{\prime}+y^{\prime}-5 x=2 t-8 \\ &y^{\prime}-8 x-y=-t^{2} \end{aligned} $$

Short answer

The general solution for the given inhomogeneous linear system is: $$ x_G(t) = c_1 \begin{pmatrix} 1 \\ 4 \end{pmatrix} e^{-t} + c_2 \begin{pmatrix} 1 \\ 2 \end{pmatrix} e^{-3t} + \begin{pmatrix} 2t+8 \\ 4t+16 \end{pmatrix} $$

Step-by-step solution

Express the system as an operator equation

Rewrite the given system in terms of the differential operator D: $$ \begin{aligned} &(D-5)x + Dy = 2t - 8 \\\ &Dy - 8x = -t^2 \end{aligned} $$

Solve for the system matrix

In order to find the system matrix, we need to take the coefficients of x and y in the operator equation. The system matrix A will be: $$ A = \begin{pmatrix} -5 & 1 \\ -8 & 1 \end{pmatrix} $$

Solve for the inhomogeneous term

Next, we need to find the inhomogeneous term \(g(t)\) for the given system. We have: $$ g(t) = \begin{pmatrix} 2t-8 \\ -t^2 \end{pmatrix} $$

Find the general solution of the homogeneous system

We need to find the general solution of the homogeneous system, which is represented as \(x(t) = c_1 x_1(t) + c_2 x_2(t)\), where \(x_1(t)\) and \(x_2(t)\) are the fundamental solutions. We first find the eigenvalues and eigenvectors of matrix A: $$ \det(A-\lambda I) = \det\begin{pmatrix} -5-\lambda & 1 \\ -8 & 1-\lambda \end{pmatrix} =(-5 - \lambda)(1 - \lambda) - (-8) = 0 $$ Solve for eigenvalues \(\lambda\): $$ \lambda^2 + 4\lambda + 3 = (\lambda + 1)(\lambda + 3) = 0 \\ \lambda_1 = -1, \hspace{1cm} \lambda_2 = -3 $$ Now find the eigenvectors corresponding to each eigenvalue: For \(\lambda_1=-1\): $$ (A - (-1)I)v_1 = \begin{pmatrix} -4 & 1 \\ -8 & 2 \end{pmatrix} v_1 = 0 \Rightarrow v_1 = \begin{pmatrix} 1 \\ 4 \end{pmatrix} $$ For \(\lambda_2=-3\): $$ (A - (-3)I)v_2 = \begin{pmatrix} -2 & 1 \\ -8 & 4 \end{pmatrix} v_2 = 0 \Rightarrow v_2 = \begin{pmatrix} 1 \\ 2 \end{pmatrix} $$ From the above results, we can construct the general solution for the homogeneous system: $$ x(t) = c_1 \begin{pmatrix} 1 \\ 4 \end{pmatrix} e^{-t} + c_2 \begin{pmatrix} 1 \\ 2 \end{pmatrix} e^{-3t} $$

Construct the general solution for the inhomogeneous system

To construct the general solution for the inhomogeneous system, we can use the formula: $$ x_G(t) = x(t) + X(t) \\ x_G(t) = c_1 \begin{pmatrix} 1 \\ 4 \end{pmatrix} e^{-t} + c_2 \begin{pmatrix} 1 \\ 2 \end{pmatrix} e^{-3t} + X(t) $$ where \(X(t)\) is the particular solution obtained from the variation of constants method: $$ X(t) = \int [A^{-1} g(t)] dt \\ X(t) = \int \begin{pmatrix} -1 & -1 \\ -4 & -2 \end{pmatrix} \begin{pmatrix} 2t-8 \\ -t^2 \end{pmatrix} dt $$ Compute the integral: $$ X(t) = \begin{pmatrix} 2t+8 - t^2 \\ 4t+16 - 2t^2 \end{pmatrix} + C \\ X(t) = \begin{pmatrix} 2t+8 \\ 4t+16 \end{pmatrix} + C $$ Finally, the general solution for the inhomogeneous system is: $$ x_G(t) = c_1 \begin{pmatrix} 1 \\ 4 \end{pmatrix} e^{-t} + c_2 \begin{pmatrix} 1 \\ 2 \end{pmatrix} e^{-3t} + \begin{pmatrix} 2t+8 \\ 4t+16 \end{pmatrix} $$

Key concepts

Linear Systems

Linear systems of differential equations are at the heart of many problems in mathematics, engineering, and the sciences. These systems are composed of multiple linear equations involving derivatives of functions. A typical linear system has a structure that can be succinctly written in matrix form as \(\mathbf{A} \mathbf{x}' + \mathbf{B} \mathbf{x} = \mathbf{g}(t)\), where \(\mathbf{A}\) and \(\mathbf{B}\) are matrices that represent the coefficients, \(\mathbf{x}\) is the vector of unknown functions, and \(\mathbf{g}(t)\) is the inhomogeneity or 'forcing' term.

Solving a system like this involves finding an expression for \(\mathbf{x}(t)\) that works for all \(t\) within a certain interval. This requires a blend of techniques that include matrix algebra, along with an understanding of differential equations and their solutions. The concept of linear systems is foundational to understanding more complex behaviors in multi-dimensional dynamic systems.

Operator Method

The operator method simplifies differential equations by using 'operators' that act upon functions. An operator is a rule that takes a function and transforms it into another function. In the context of differential equations, the differential operator \(D\) plays a central role, where \(Dy\) represents the derivative \(y'\) of the function \(y\) with respect to the independent variable, usually time \(t\).

In our exercise, the differential operator \(D\) was used to rewrite the system of equations in a more manageable form. It's a powerful technique that can transform a complex system into a format that's easier to analyze and solve. Understanding how to manipulate and apply these operators to both sides of an equation can reveal solutions that may not be immediately apparent.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are crucial concepts when dealing with linear differential equations and linear algebra. An eigenvalue of a matrix \(\mathbf{A}\) is a scalar \(\lambda\) such that there exists a non-zero vector (the eigenvector) which, when multiplied by \(\mathbf{A}\), is simply scaled by \(\lambda\). This can be expressed as \(\mathbf{A}\mathbf{v} = \lambda\mathbf{v}\).

In the context of solving linear systems, the eigenvalues give us the 'growth rates' of the solutions, while the eigenvectors give us the 'directions' along which these solutions grow or decay. Identifying these eigenvalues and eigenvectors allows us to construct the fundamental solutions to a homogenous system, providing key building blocks for a general solution.

Inhomogeneous System

An inhomogeneous system of differential equations is one that includes a non-zero forcing term, represented by \(\mathbf{g}(t)\) in matrix notation. Unlike homogeneous systems, where \(\mathbf{g}(t) = \mathbf{0}\), inhomogeneous systems cannot be solved solely with the theory of eigenvalues and eigenvectors.

These systems often model real-world scenarios where some external influence is at play, such as a driving force, a source term, or an input signal. Finding the particular solution \(X(t)\) that satisfies the inhomogeneous part is a crucial step in constructing the general solution for these kinds of systems. The exercise we’re reviewing demonstrates the process of obtaining such a solution using techniques suited for inhomogeneous systems.

Variation of Constants Method

The variation of constants method is a technique used to find a particular solution to an inhomogeneous linear differential equation. This method assumes that the constants in the general solution of the related homogeneous equation can be replaced with functions that vary over time, hence the name. These functions are then determined by solving additional equations that arise from plugging the variable 'constants' back into the original inhomogeneous equation.

In our exercise, after constructing the solution to the homogeneous system, the variation of constants method would involve integrating the inverse of the matrix \(\mathbf{A}\) multiplied by the inhomogeneity \(\mathbf{g}(t)\) to find the particular solution \(\mathbf{X}(t)\). By understanding and applying this method, one can address a wide range of inhomogeneous systems beyond what straightforward eigenvalue-eigenvector methods can accommodate.