Question

Verify that the given vector is the general solution of the corresponding homogeneous system, and then solve the non-homogeneous system. Assume that \(t>0 .\) $$ t \mathrm{x}^{\prime}=\left(\begin{array}{cc}{3} & {-2} \\ {2} & {-2}\end{array}\right) \mathrm{x}+\left(\begin{array}{c}{-2 t} \\\ {t^{4}-1}\end{array}\right), \quad \mathbf{x}^{(c)}=c_{1}\left(\begin{array}{c}{1} \\ {2}\end{array}\right) t^{-1}+c_{2}\left(\begin{array}{c}{2} \\ {1}\end{array}\right) t^{2} $$

Short answer

#tag_title# Step 2: Find the Particular Solution for the Non-Homogeneous System #tag_content# For the non-homogeneous system, we can use an ansatz method. Since the inhomogeneity term is a polynomial of degree 1, \(\mathbf{f}(t) = \begin{pmatrix}t \\ t^2\end{pmatrix}\), we will guess a particular solution of the form \(\mathbf{x}^{(p)} = \begin{pmatrix}at \\ bt^2\end{pmatrix}\), where a and b are constants to be determined. Find the derivative \(\mathbf{x}^{(p)\prime}\) and plug in both \(\mathbf{x}^{(p)}\) and \(\mathbf{x}^{(p)\prime}\) into the non-homogeneous system. Then, solve for a and b. #tag_title# Step 3: Write Down the General Solution of the Non-Homogeneous System #tag_content# The general solution of the non-homogeneous system is \(\mathbf{x}(t) = \mathbf{x}^{(h)} + \mathbf{x}^{(p)}\). Substitute the expressions for \(\mathbf{x}^{(h)}\) and \(\mathbf{x}^{(p)}\) and write the general solution in the form \(\mathbf{x}(t) = \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix}\). Based on the given solution, write a short answer question that prompts students to verify if the given vector is a solution to the homogeneous system, find the particular solution for the non-homogeneous system, and then write down the general solution of the non-homogeneous system. Question: Given the homogeneous system \(t\mathbf{x}^{\prime} = A\mathbf{x}\), where A is a matrix \(\begin{pmatrix} 3 & -2 \\ 2 & -2\end{pmatrix}\), and a general solution for the homogeneous system \(\mathbf{x}^{(h)} = c_1\begin{pmatrix} 1 \\ 2\end{pmatrix}t^{-1} + c_2\begin{pmatrix} 2 \\ 1\end{pmatrix}t^{2}\), verify if the given vector is a solution to the homogeneous system. Then, consider the non-homogeneous system with an inhomogeneity term \(\mathbf{f}(t) = \begin{pmatrix}t \\ t^2\end{pmatrix}\), and find the particular solution. Lastly, write down the general solution of the non-homogeneous system.

Step-by-step solution

Verify the General Solution for the Homogeneous System

Let \(\mathbf{x}^{(h)} = c_1\begin{pmatrix} 1 \\ 2\end{pmatrix}t^{-1} + c_2\begin{pmatrix} 2 \\ 1\end{pmatrix}t^{2}\). First, we need to find the derivative \(\mathbf{x}^{(h)\prime}\). Then, plug in both \(\mathbf{x}^{(h)}\) and \(\mathbf{x}^{(h)\prime}\) into the homogeneous system \(t\mathbf{x}^{\prime} = A\mathbf{x}\), where A is a matrix \(\begin{pmatrix} 3 & -2 \\ 2 & -2\end{pmatrix}\).

Key concepts

Homogeneous Systems

Homogeneous systems of differential equations are foundational in understanding how combinations of growth and decay processes interact over time. A system is called homogeneous if it can be expressed in the form \(t\mathbf{x}^{\prime} = A\mathbf{x}\), where A is a constant matrix and \(\mathbf{x}\) is a vector of unknown functions. It's important to note that there is no external forcing term; the system evolves based on its initial conditions alone. To verify a general solution for a homogeneous system, like \(\mathbf{x}^{(h)}\) in our exercise, one must check if it satisfies the standard form equation when substituting \(\mathbf{x}\) and its derivative \(\mathbf{x}^{\prime}\).

The process involves finding the derivative of the proposed solution, plugging both the original solution and its derivative back into the original equation, and ensuring that both sides of the equation match. In essence, the verification confirms that the proposed solution indeed describes the behavior of the system over time, without the influence of outside forces.

Non-Homogeneous Systems

In contrast, non-homogeneous systems include an additional term, often related to external influences or inputs. The general form is \(t\mathbf{x}^{\prime} = A\mathbf{x} + \mathbf{f}(t)\), where \(\mathbf{f}(t)\) is a non-zero vector function of time. This term makes the system non-homogeneous because the system's evolution now depends on factors beyond the initial conditions and the natural dynamics dictated by matrix A. Solving such a system typically involves two main steps: finding the general solution to the associated homogeneous system (which is what we've called \(\mathbf{x}^{(c)}\) in our exercise) and then finding a particular solution that accounts for \(\mathbf{f}(t)\).

The superposition principle allows us to add the particular solution to the general solution of the homogeneous system to produce the general solution to the non-homogeneous system. This is powerful because it tells us that once we have a complete solution set for the homogeneous case, we only need one additional solution to account for the non-homogeneous part.

Matrix Differential Equations

Matrix differential equations encapsulate systems of first-order ordinary differential equations into a compact form. They boast not only a more elegant representation but also allow for a systematic approach to finding solutions through linear algebra techniques. The equation \(t\mathbf{x}^{\prime} = A\mathbf{x} + \mathbf{f}(t)\) from our previous section is an example of a matrix differential equation. Here, the matrix A represents the coefficients of the unknown functions in vector \(\mathbf{x}\), while \(\mathbf{x}^{\prime}\) signifies the vector of derivatives.

The beauty lies in dealing with only one matrix equation instead of several scalar differential equations. Analytical solutions might involve eigenvalues and eigenvectors of the matrix A, which correspond to the system's modes of independent behavior. The matrix approach is particularly handy when dealing with higher-dimensional systems where scalar equations become cumbersome.

General Solution

The general solution to a system of differential equations encompasses all possible solutions and is composed of two parts when dealing with non-homogeneous systems. For homogeneous systems, it's a combination of solutions based on the system's characteristic equation, often involving constants like \(c_1, c_2, ..., c_n\) that can be adjusted to fit specific initial conditions. In the non-homogeneous case, as seen in our exercise, the general solution also includes a particular solution that specifically satisfies the non-zero term \(\mathbf{f}(t)\).

When confirming a general solution, one must verify that it not only fulfills the associated homogeneous equation but also renders the entire non-homogeneous equation valid. This process underscores the importance of understanding both the homogeneous and particular components, as their combination holds the key to unlocking the complete description of the system's dynamics over time.