Question

Prove that for the system $$ d x / d t=F(x, y), \quad d y / d t=G(x, y) $$ there is at most one trajectory passing through a given point \(\left(x_{0}, y_{0}\right)\) Hint: Let \(C_{0}\) be the trajectory generated by the solution \(x=\phi_{0}(t), y=\psi_{0}(t),\) with \(\phi_{0}\left(l_{0}\right)=\) \(x_{0}, \psi_{0}\left(t_{0}\right)=y_{0},\) and let \(C_{1}\) be trajectory generated by the solution \(x=\phi_{1}(t), y=\psi_{1}(t)\) with \(\phi_{1}\left(t_{1}\right)=x_{0}, \psi_{1}\left(t_{1}\right)=y_{0}\). Use the fact that the system is autonomous and also the existence and uniqueness theorem to show that \(C_{0}\) and \(C_{1}\) are the same.

Short answer

Short Answer: There can be at most one trajectory passing through a given point $(x_0, y_0)$ in an autonomous system because the Existence and Uniqueness theorem guarantees a unique solution for the system of equations with a given initial condition. By considering two trajectories passing through $(x_0, y_0)$ and comparing their behavior in different time frames, we can conclude that they are the same, thus proving that there is at most one trajectory passing through a given point.

Step-by-step solution

Consider two trajectories \(C_0\) and \(C_1\)

Let's assume that there are two trajectories \(C_0\) and \(C_1\) through a given point \((x_0, y_0)\). For the trajectory \(C_0\), we have \(x=\phi_0(t)\) and \(y=\psi_0(t)\), with the initial condition \(\phi_0(t_0)=x_0\) and \(\psi_0(t_0)=y_0\). Similarly, for trajectory \(C_1\), we have \(x=\phi_1(t)\) and \(y=\psi_1(t)\), with the initial condition \(\phi_1(t_1)=x_0\) and \(\psi_1(t_1)=y_0\).

Apply the Existence and Uniqueness theorem

The Existence and Uniqueness theorem guarantees that for a given initial condition \((x_0, y_0)\), there exists a unique solution for the system of equations. So, both pairs of functions \(\phi_0(t)\), \(\psi_0(t)\) and \(\phi_1(t)\), \(\psi_1(t)\) are representing the unique solution of the given system that passes through \((x_0, y_0)\) at two different times, \(t_0\) and \(t_1\), respectively.

Use the autonomy of the system to compare trajectories

An autonomous system is one where the functions \(F(x, y)\) and \(G(x, y)\) don't explicitly depend on the time variable \(t\). Therefore, the solutions \(\phi_0(t)\), \(\psi_0(t)\) and \(\phi_1(t)\), \(\psi_1(t)\) should have the same qualitative behavior and shape, regardless of the time frame we're analyzing. To compare the trajectories, we can create a new function by changing the time variable: - Consider the functions \(\phi_1(t - (t_1 - t_0))\) and \(\psi_1(t - (t_1 - t_0))\) - In this new time frame, these functions should now have the initial conditions \(\phi_1(0 - (t_1 - t_0)) = \phi_1(t_0 - t_1) = x_0\) and \(\psi_1(0 - (t_1 - t_0)) = \psi_1(t_0 - t_1) = y_0\) - Since the original functions \(\phi_0(t)\) and \(\psi_0(t)\) also have the initial conditions \(\phi_0(t_0) = x_0\) and \(\psi_0(t_0)=y_0\), we now have two solutions of the same system with the same initial conditions

Conclude that the trajectories are the same

Applying the Existence and Uniqueness theorem again, since both the new time frame functions \(\phi_1(t - (t_1 - t_0))\) and \(\psi_1(t - (t_1 - t_0))\) and the original functions \(\phi_0(t)\) and \(\psi_0(t)\) have the same initial conditions, they must represent the same trajectory. Therefore, we can conclude that the trajectories \(C_0\) and \(C_1\) are the same, proving that there is at most one trajectory passing through a given point \((x_0, y_0)\).

Key concepts

Autonomous System

An autonomous system in the context of differential equations is one where the equations do not explicitly depend on the independent variable, usually time. This means that the system evolves based on its current state alone, not on when an observation is made. The form of autonomous systems is typically:

• \( \frac{dx}{dt} = F(x, y) \)
• \( \frac{dy}{dt} = G(x, y) \)

Here, \(F\) and \(G\) are functions of the variables \(x\) and \(y\), but not of \(t\), showcasing the characteristic autonomy. The key feature of autonomous systems is that their behavior is consistent over time. No matter when the system is analyzed, it will behave in the same way under the same conditions. This property of consistency is vital when computing trajectories, making autonomous systems easier to analyze in many scenarios where time independence is assumed.

Differential Equations

Differential equations are equations that relate a function to its derivatives. They are fundamental in expressing physical phenomena and systems across disciplines like physics, engineering, biology, and economics. The given system of differential equations:

• \( \frac{dx}{dt} = F(x, y) \)
• \( \frac{dy}{dt} = G(x, y) \)

represents a system where changes in \(x\) and \(y\) are determined by the functions \(F\) and \(G\). The solutions to these equations are functions, \(x(t)\) and \(y(t)\), which describe how \(x\) and \(y\) evolve over time. Solving a differential equation typically involves finding these functions given some initial conditions. This evolution can be visualized as a trajectory in a plane, which changes direction according to the rules set by \(F\) and \(G\). Understanding how to approach and solve these equations is crucial for applying mathematical models to real-world problems.

Initial Condition

Initial conditions in differential equations specify the state of the system at the beginning of the analysis. They are necessary for finding a unique solution to the differential system. For the given system of equations, the initial conditions can be expressed as:

• \( \phi_0(t_0) = x_0 \)
• \( \psi_0(t_0) = y_0 \)

This means at time \(t_0\), the values of \(x\) and \(y\) are \(x_0\) and \(y_0\), respectively. Initial conditions are critical because the Existence and Uniqueness Theorem heavily relies on them to guarantee the uniqueness of the solution. This theorem states that if certain conditions are met, including the specification of initial conditions, then there will exist a unique solution trajectory starting from that initial point. By providing the values of \(x\) and \(y\) at an initial time, the trajectory of this system’s state through time can be precisely determined.

Trajectory

A trajectory in the context of differential equations is the path traced by the solution in its phase space as it evolves over time. Think of it as the 'road' taken by the system's state — represented by \(x(t)\) and \(y(t)\) in the given system — as it progresses according to the rules of the differential equations. For the problem:

• \( \phi_0(t) \), \( \psi_0(t) \) for trajectory \(C_0\)
• \( \phi_1(t) \), \( \psi_1(t) \) for trajectory \(C_1\)

Each set of solutions determines a trajectory through the \((x, y)\) plane. However, the Existence and Uniqueness Theorem ensures that for each initial point, there is only one trajectory. This means if we begin at a particular \((x_0, y_0)\), all solutions will trace the same path, negating the possibility of multiple trajectories through that point. Thus, trajectories provide a geometric interpretation of differential solutions and are invaluable in visualizing and understanding the system's dynamics.