Question

The system \(\left.t \mathbf{x}^{\prime}=\mathbf{A} \mathbf{x} \text { is analogous to the second order Euler equation (Section } 5.5\right) .\) Assum- ing that \(\mathbf{x}=\xi t^{\prime},\) where \(\xi\) is a constant vector, show that \(\xi\) and \(r\) must satisfy \((\mathbf{A} \mathbf{I}) \boldsymbol{\xi}=\mathbf{0}\) in order to obtain nontrivial solutions of the given differential equation.

Short answer

Answer: \((\mathbf{A} - r\mathbf{I})\boldsymbol{\xi}=\mathbf{0}\).

Step-by-step solution

Find x'

To begin, we need to differentiate the given \(\mathbf{x}\) expression with respect to "t", using the assumption that \(\mathbf{x} = \xi t^{r}\). \(\mathbf{x}^{\prime}(t) = \frac{d (\xi t^{r})}{dt} = r \cdot \xi \cdot t^{r-1}\)

Replace x and x' in the given system

Now we can plug \(\mathbf{x}^{\prime}\) and \(\mathbf{x}\) into the given system: \(t \mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}\). \(t(r \cdot \xi \cdot t^{r-1}) = \mathbf{A} (\xi t^{r})\)

Rewrite and Simplify the Equation

We can now simplify and rewrite the equation to achieve our goal. \(r \cdot \xi \cdot t^{r} = \mathbf{A} \cdot \xi \cdot t^{r}\) Divide both sides of the equation by \(t^{r}\): \(r \cdot \xi = \mathbf{A} \cdot \xi\) Now rewrite the equation in terms of \((\mathbf{A} - r\mathbf{I}) \boldsymbol{\xi}\): \((\mathbf{A} - r\mathbf{I}) \boldsymbol{\xi} = \mathbf{0}\) We have shown that to obtain nontrivial solutions of the given differential equation, \(\xi\) and \(r\) must satisfy our final equation \((\mathbf{A} - r\mathbf{I})\boldsymbol{\xi}=\mathbf{0}\).

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are fundamental in expressing the dynamics of various systems, from simple mechanical motions to complex biological processes.

In the context of Euler equations, we're dealing with a specific type of differential equation where the variable, in this case time ‘t’, is also present in the coefficients. The Euler equation represents systems where the rates of change are proportional to the function itself, multiplied by a power of the independent variable. Such equations often arise in problems involving growth and decay, fluid flow, or in the oscillations of springs and electric circuits.

Solving differential equations typically involves finding a function or set of functions that satisfy the given equation. In the example provided, we are dealing with a system of differential equations where the solution is assumed to have a particular form involving the constant vector \(\xi\) and a power of 't'. Understanding how to manipulate these equations to find solutions is essential for students aiming to master the dynamics of various physical systems.

Nontrivial Solutions

In the study of differential equations, a nontrivial solution is one that is more meaningful than the trivial solution, which often implies the function being equal to zero everywhere. Nontrivial solutions contain information about the behavior of the system under study and are often sought after in engineering and physics problems where the zero solution would offer no insight.

Normally, multiple solutions to a differential equation can exist, including both trivial and nontrivial ones. Identifying conditions under which nontrivial solutions can be obtained is important in characterizing the system's behavior. In the Euler equation exercise, we are looking for nontrivial solutions by setting up an equation \( (\mathbf{A} - r\mathbf{I}) \boldsymbol{\xi} = \mathbf{0} \) where \( \xi \) is not the zero vector. This equation is vital since it helps us find the relationships between the eigenvalues \( r \) and eigenvectors \( \xi \) of the matrix \( \mathbf{A} \) that will give us the nontrivial solutions we are interested in.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are concepts in linear algebra that play a crucial role in the analysis of linear transformations. An eigenvector of a square matrix \( \mathbf{A} \) is a nonzero vector \( \xi \) that, when multiplied by \( \mathbf{A} \) results in a scalar multiple of itself. The scalar is known as an eigenvalue, and it is denoted by \( r \) in the context of the exercise.

When we consider the equation \( (\mathbf{A} - r\mathbf{I}) \boldsymbol{\xi} = \mathbf{0} \), an eigenvalue \( r \) is a number for which this equation has nontrivial solutions for \( \xi \). These solutions are the corresponding eigenvectors. In physical systems, eigenvalues can represent natural frequencies of the system, and eigenvectors can represent the modes of vibration.

Understanding how to determine eigenvalues and eigenvectors is crucial when solving systems of differential equations. They provide the key to simplifying complex systems into more manageable ones, allowing us to predict and understand system behavior and stability. Eigenvalues and eigenvectors also have wide applications in other fields such as quantum mechanics, stability analysis, and in algorithms for machine learning and data analysis.