Question

Verify that the given vector satisfies the given differential equation. \(\mathbf{x}^{\prime}=\left(\begin{array}{ll}{3} & {-2} \\ {2} & {-2}\end{array}\right) \mathbf{x}, \quad \mathbf{x}=\left(\begin{array}{l}{4} \\\ {2}\end{array}\right) e^{2 t}\)

Short answer

Answer: Yes, the given vector function \(\mathbf{x}\) satisfies the given differential equation.

Step-by-step solution

Compute the derivative of \(\mathbf{x}\)

First, take the derivative of the given vector function \(\mathbf{x}\) with respect to \(t\): \(\frac{d}{dt}\begin{pmatrix} 4e^{2t} \\ 2e^{2t} \end{pmatrix} = \begin{pmatrix} 4(2e^{2t}) \\ 2(2e^{2t}) \end{pmatrix} = \begin{pmatrix} 8e^{2t} \\ 4e^{2t} \end{pmatrix}.\)

Multiply the matrix A by the given vector function \(\mathbf{x}\)

Next, multiply the given matrix \(A\) by the vector function \(\mathbf{x}\): \(\begin{pmatrix} 3 & -2 \\ 2 & -2 \end{pmatrix}\begin{pmatrix} 4e^{2t} \\ 2e^{2t} \end{pmatrix} = \begin{pmatrix} 3(4e^{2t}) - 2(2e^{2t}) \\ 2(4e^{2t}) - 2(2e^{2t}) \end{pmatrix} = \begin{pmatrix} 8e^{2t} \\ 4e^{2t} \end{pmatrix}.\)

Compare the results

Finally, compare the results obtained in steps 1 and 2 to check if they are equal: \(\frac{d}{dt}\mathbf{x} = \begin{pmatrix} 8e^{2t} \\ 4e^{2t} \end{pmatrix} = A\mathbf{x} = \begin{pmatrix} 8e^{2t} \\ 4e^{2t} \end{pmatrix}.\) Since the two results are equal, the given vector function \(\mathbf{x}\) satisfies the given differential equation.

Key concepts

Eigenvalues and Eigenvectors

Understanding eigenvalues and eigenvectors is essential in the study of differential equations, as they often reveal the underlying structure of the system being analyzed.

Eigenvalues, denoted typically by \(\lambda\), are scalars associated with a linear system of equations. They are special values for which there exists a non-zero vector \(\mathbf{v}\) such that when the system matrix, let's call it \(A\), multiplies this vector, the output is a scalar multiple of \(\mathbf{v}\), expressed as \(A\mathbf{v} = \lambda\mathbf{v}\). The vector \(\mathbf{v}\) associated with \(\lambda\) is known as an eigenvector. Together, they play a pivotal role in simplifying matrix operations and solving systems of differential equations.

For instance, in the given exercise, an understanding of eigenvalues and eigenvectors could potentially offer a more profound insight into why the vector \(\mathbf{x}\) behaves the way it does when multiplied by the matrix. In practical terms, they can dramatically simplify difficult differential equations by decoupling the system into independent equations, each corresponding to an eigenvector direction.

Matrix Multiplication

Matrix multiplication is a cornerstone of linear algebra. It is a procedure by which two matrices are combined to form a new matrix. The multiplication process involves taking the dot product of rows from the first matrix with columns from the second matrix.

In the context of the differential equation problem presented, the operation involves multiplying a 2x2 matrix by a 2x1 vector. The outcome is a new 2x1 vector. Symbolically, for given matrix \(A\) and vector \(\mathbf{x}\), we compute \(A\mathbf{x}\) to get Another vector of the same dimensions as \(\mathbf{x}\).

During this multiplication, each entry of the resulting vector is a sum of products: for example, if \(A\) is a matrix with entries \(a_{ij}\) and \(\mathbf{x}\) is a vector with entries \(x_i\), then the first entry of the resulting vector is \(a_{11}x_1 + a_{12}x_2\), and so on for the remaining entries. This operation is fundamental in many aspects of applied mathematics, including the analysis of linear transformations and systems of linear equations.

Exponential Functions

Exponential functions are a category of mathematical functions denoted by the form \(f(t) = ae^{kt}\), where \(e\) is the base of the natural logarithm, and \(a\) and \(k\) are constants. These functions are characterized by their rate of growth or decay, which is dictated by the constant \(k\).

In differential equations, exponential functions come into play notably when dealing with solutions that involve growth and decay processes, oscillations, and other phenomena that can be described by rates of change proportional to the function's value.

The exercise provided involves an exponential function, where \(e^{2t}\) denotes an exponential growth correlated with time \(t\). The significance of the exponential function in this context lies in how it describes the behavior of \(\mathbf{x}\) under the linear transformation dictated by the differential equation. Specifically, the solution given demonstrates that regardless of the time \(t\), the relationship defined by the differential equation holds true—a hallmark of exponential functions.