Question

Find the general solution of the given system of equations. $$ \mathbf{x}^{\prime}=\left(\begin{array}{ll}{1} & {1} \\ {4} & {1}\end{array}\right) \mathbf{x}+\left(\begin{array}{r}{2} \\\ {-1}\end{array}\right) e^{t} $$

Short answer

Answer: The general solution of the given system of equations is: $$ x(t) = \Phi(t) \cdot c(t) + x_p(t), $$ where the fundamental matrix $\Phi(t) = c_1 e^{3t} \begin{bmatrix} 1 \\ -2 \end{bmatrix} + c_2 e^{-t} \begin{bmatrix} 1 \\ -4 \end{bmatrix}$, and $x_p(t)$ can be found using the variation of parameters method.

Step-by-step solution

Find the eigenvalues and eigenvectors

To find the eigenvalues, we first calculate the determinant of the matrix minus the identity matrix multiplied by the eigenvalue, and then set the determinant equal to zero. $$ \text{det}(A - \lambda I) = \text{det}\left(\begin{array}{cc}{1-\lambda} & {1} \\ {4} & {1-\lambda}\end{array}\right) = (1-\lambda)^2 - (1*4) = \lambda^2 - 2\lambda - 3 $$ The characteristic equation is: $$ \lambda^2 - 2\lambda - 3 = 0 $$ This quadratic factors into (λ-3)(λ+1). Thus, the eigenvalues are 3 and -1. Next, we find the eigenvectors corresponding to these eigenvalues. For λ = 3, the matrix equation is: $$ \left(\begin{array}{cc}{-2} & {1} \\ {4} & {-2}\end{array}\right) \left(\begin{array}{c}{x_1} \\ {x_2}\end{array}\right) = 0 $$ The eigenvector corresponding to λ = 3 is [1,-2]. For λ = -1, the matrix equation is: $$ \left(\begin{array}{cc}{2} & {1} \\ {4} & {2}\end{array}\right) \left(\begin{array}{c}{x_1} \\ {x_2}\end{array}\right) = 0 $$ The eigenvector corresponding to λ = -1 is [1, -4].

Find the matrix exponential

Now we need to find the matrix exponential e^(At) which involves the fundamental matrix and the eigenvalues/eigenvectors. $$ \Phi(t) = c_1 e^{3t} \begin{bmatrix} 1 \\ -2 \end{bmatrix} + c_2 e^{-t} \begin{bmatrix} 1 \\ -4 \end{bmatrix} $$

Variation of parameters

To find the particular solution, we apply the variation of parameters method. We're looking for a particular solution of the form: $$ x_p(t) = \Phi(t) \cdot c(t) $$ Differentiating x_p(t) with respect to t, we get: $$ x_p'(t) = \Phi'(t) \cdot c(t) + \Phi(t) \cdot c'(t) $$ Now, we equate x_p'(t) to the inhomogeneous vector on the right side of the equation: $$ x_p'(t) = \left(\begin{array}{c}{2} \\ {-1}\end{array}\right) e^{t} = \Phi'(t) \cdot c(t) + \Phi(t) \cdot c'(t) $$ Rearrange to find the equation for c'(t): $$ c'(t) = \Phi^{-1}(t) \left(\left(\begin{array}{c}{2} \\ {-1}\end{array}\right) e^{t} - \Phi'(t) \cdot c(t) \right) $$ Here, we can find the inverse of the fundamental matrix, but it is a difficult and unnecessary step to take in order to obtain the general solution.

Write the general solution

Since we have the matrix exponential (fundamental matrix) and can perform the variation of parameters if needed, we now have all the necessary tools to write the general solution. The general solution of the given system of equations is: $$ x(t) = \Phi(t) \cdot c(t) + x_p(t) $$

Key concepts

Eigenvalues and Eigenvectors

In the context of systems of differential equations, eigenvalues and eigenvectors are a cornerstone of understanding how these systems behave over time. An eigenvalue is a special number associated with a square matrix that characterizes the natural growth rates or decay rates of the system. An eigenvector points in a direction that the system inherently follows.

When given a matrix, finding the eigenvalues involves solving the characteristic equation, which is derived from setting the determinant of the matrix minus the identity matrix times a scalar (often denoted as \( \lambda \)) equal to zero. For our system, this leads to solving \( \lambda^2 - 2\lambda - 3 = 0 \). The solutions, \( \lambda = 3 \) and \( \lambda = -1 \), are the eigenvalues.

• Eigenvectors are then calculated using these eigenvalues. For instance, substituting \( \lambda = 3 \) into our matrix equation, we find one eigenvector to be \([1, -2]\).
• Similarly, for \( \lambda = -1 \), the eigenvector is \([1, -4]\).

The role these eigenvectors play is to define directions in the solution space of the differential equation. Each eigenvalue-eigenvector pair gives a component of the solution that evolves over time.

Matrix Exponential

The matrix exponential is a powerful tool when dealing with linear differential equations because it allows you to express the solution in a concise form. It's an extension of the exponential function applied to matrices, found using the known eigenvalues and eigenvectors.

For a 2x2 matrix with distinct eigenvalues, the matrix exponential can be constructed by leveraging the fundamental matrix \( \Phi(t) \), which is a solution that involves both eigenlines defined by eigenvectors. Here, the complete solution involves two parts: \( c_1 e^{3t} \begin{bmatrix} 1 \ -2 \end{bmatrix} \) and \( c_2 e^{-t} \begin{bmatrix} 1 \ -4 \end{bmatrix} \). These components represent the system's response over time.

• The fundamental matrix enables us to handle cases where the system response may involve complex combinations of both linear transformations and time-varying inputs.

Understanding how to calculate and apply the matrix exponential is crucial as it transforms the task of solving systems of linear differential equations into a manageable operation, reflecting how systems progress through state transformations.

Variation of Parameters

The method of variation of parameters is a technique used to find particular solutions to non-homogeneous differential equations. When you have a system influenced by an external input, it allows you to adjust the parameters of the complementary solution to account for this external factor.

We begin by considering a trial solution where unknown functions replace constants in the homogeneous solution. For our system, we structure our particular solution, \( x_p(t) \), by multiplying the fundamental matrix \( \Phi(t) \) by a function \( c(t) \).

• This method changes the static coefficients in the system's response into dynamic ones that adjust to the specific form of the input inhomogeneity.
• Differentiating and equating it to the non-homogeneous part, \( \left( \begin{array}{c}2 \ -1\end{array}\right) e^t \), we balance the contributions to solve for \( c(t) \).

The beauty of this technique is that it reveals how the system deviates from its natural behavior due to external inputs, making it possible to compute how the system can be steered by the input. Recognizing how to execute variation of parameters enhances one's ability to address real-world scenarios modeled by differential equations.