Question

Prove that any solution of an equation \(\dot{x}=\boldsymbol{v}(t, x)\) defined by a direction field in \(R \times R^{\text {" can be extended indefinitely if } v \text { grows no faster than the first power }} \end{array}\) of \(x\) at infinity, i.e., if \(|v(t, x)| \leq k|x|\) for all \(t\) and all \(|x| \geq r\), where \(r\) and \(k\) are constants.

Short answer

Short Answer: Given a first-order differential equation \(\dot{x} = \boldsymbol{v}(t,x)\) with the condition \(|v(t, x)| \leq k|x|\) for all \(t\) and all \(|x| \geq r\), we showed that the solution curve \(x(t)\) remains bounded for all t by demonstrating that the change in magnitude of \(x(t)\) is no larger than the integral of a linear function of \(x(t)\). Since the solution curve remains bounded, it can be extended indefinitely, proving the statement.

Step-by-step solution

Restate the given information

We are given a first-order differential equation of the form: \(\dot{x} = \boldsymbol{v}(t,x)\), where \(\boldsymbol{v}(t,x)\) is a direction field in \(R \times R^n\). It is stated that \(|v(t, x)| \leq k|x|\) for all \(t\) and all \(|x| \geq r\), where \(r\) and \(k\) are constants.

Introduce the solution curve

Let \(x(t)\) be a solution curve of the given differential equation. Since \(\dot{x} = \boldsymbol{v}(t, x(t))\), we want to show that this solution curve can be extended indefinitely if the given condition holds.

Show the solution remains bounded for all t

Observe that the given condition \(|v(t, x)| \leq k|x|\) states that the rate of change of \(x(t)\) is no larger than a linear function of \(x(t)\). For \(|x(t)| \geq r\), we have: \(|\dot{x}| = |\boldsymbol{v}(t, x(t))| \leq k|x(t)|\) Now, let us integrate both sides of this inequality with respect to t, over the interval \([t_0, t_1]\): \(\int_{t_0}^{t_1} |\dot{x}| dt \leq k\int_{t_0}^{t_1}|x(t)| dt\) Recall that \(\int_{t_0}^{t_1} |\dot{x}| dt\) represents the change in magnitude of \(x(t)\) from \(t_0\) to \(t_1\). Therefore, we obtain: \(\Delta|x| \leq k\int_{t_0}^{t_1}|x(t)| dt\) Thus, the change in magnitude of \(x(t)\) is no larger than the integral of a linear function of \(x(t)\). This means that the solution curve \(x(t)\) remains bounded for all t.

Extend the solution indefinitely

Since we have shown that the solution curve \(x(t)\) remains bounded for all t, it follows that \(x(t)\) can be extended indefinitely. This concludes the proof that any solution of the given first-order differential equation can be extended indefinitely if the rate of growth of the vector field is no faster than the first power of x at infinity.

Key concepts

Direction Fields

When working with differential equations, a direction field is a powerful tool to visualize solutions. It provides a graphical representation indicating the direction in which the solution curves move at any given point in the plane. By plotting small arrows or line segments at various points, each representing the slope given by \(\dot{x} = \boldsymbol{v}(t, x)\), one can get an intuitive sense of the behavior of solutions.

This visualization helps to identify patterns and predict the behavior of solutions, including whether they tend to increase, decrease, or oscillate. It also allows you to anticipate how solutions might extend over time or under certain conditions.

Direction fields serve as a preliminary step in analyzing differential equations, helping students to understand the dynamics without needing to solve the equation analytically first.

Solution Curves

Solution curves, also known as integral curves or trajectories, represent the solutions of a differential equation in a direction field. They are the paths traced by solutions through the space defined by the equations. For the differential equation \(\dot{x} = \boldsymbol{v}(t, x)\), each solution curve corresponds to specific initial conditions and satisfies the equation over time.

These curves illustrate how a solution evolves and can help determine whether solutions converge to a point, diverge, or form a periodic pattern.

• Provide a visual trace of where the solutions go over time.
• Help in understanding the stability and nature of equilibrium points.
• Useful to deduce global behavior from local information.

Understanding solution curves allows students to transform abstract mathematical equations into interpretable paths, clarifying the dynamic systems being modeled.

Bounded Functions

A function is said to be bounded if its values do not escape within a specified boundary, often defined mathematically as values that do not exceed a particular magnitude. In the context of the given differential equation, \(\dot{x} = \boldsymbol{v}(t, x)\), a bounded solution curve is one where the magnitude of \(x(t)\) stays within certain limits over time, provided the vector field \(\boldsymbol{v}\) satisfies the condition \(|v(t, x)| \leq k|x|\).

Such a condition often ensures that solutions do not spiral out to infinity uncontrollably. Boundedness is crucial for determining the endless extension of solution curves.

• It restricts the growth of solutions, ensuring they remain contained.
• Essential for proving the long-term behavior of solutions.
• Critical for confirming that solutions can be meaningfully extended indefinitely.

In essence, boundedness provides assurance of the manageability and predictability of the solutions throughout the interval of interest.

First-order Differential Equations

First-order differential equations involve the first derivative of a function and provide a foundational understanding of dynamic systems. These equations represent rates of change and are pivotal in modeling real-world systems ranging from populations in biology to circuits in engineering. The standard form given in the exercise, \(\dot{x} = \boldsymbol{v}(t, x)\), shows a simple yet profound relationship where the rate \(\dot{x}\) depends on both the current state and time.

Key characteristics include:

• Representing dynamic processes and changes over time.
• Incorporating both linear and nonlinear relationships.
• Facilitating the study of systems' stability and behavior.

Understanding first-order differential equations is crucial as they serve as building blocks for more complex multi-variable systems. They allow us to study how quantities evolve, making them invaluable in predicting future system states under given conditions.