Question

(a) Find an equation of the form \(H(x, y)=c\) satisfied by the trajectories. (b) Plot several level curves of the function \(H\). These are trajectories of the given system. Indicate the direction of motion on each trajectory. $$ d x / d t=2 y, \quad d y / d t=8 x $$

Short answer

Answer: No, we cannot find an equation representing the trajectories of the given system of differential equations.

Step-by-step solution

Find the first integral H(x,y)

Given the system of differential equations \(\frac{\mathrm{d}x}{\mathrm{d}t} = 2y\) and \(\frac{\mathrm{d}y}{\mathrm{d}t} = 8x\), we need to find a first integral \(H(x, y)\) such that \(\frac{\partial H}{\partial x} = 2y\) and \(\frac{\partial H}{\partial y} = 8x\). To find \(H(x, y)\), we can integrate both equations with respect to \(x\) and \(y\): By integrating \(\frac{\partial H}{\partial x} = 2y\) with respect to \(x\) we get: $$ H(x,y) = 2xy + g(y) $$ where g(y) is an arbitrary function of \(y\). Now, we differentiate \(H(x, y)\) with respect to \(y\): $$ \frac{\partial H}{\partial y} = 2x + g'(y) $$ Since \(\frac{\partial H}{\partial y} = 8x\), we can write: $$ 2x + g'(y) = 8x $$ By comparing the coefficients of \(x\), we find: $$ g'(y) = -6x $$ However, we expect \(g'(y)\) to be a function of \(y\) alone. The given system of differential equations does not allow for such a function \(H(x,y)\). So we cannot derive an equation of the form \(H(x,y) = c\) for the trajectories in this case.

Plot level curves and indicate the direction of motion

Since we cannot find a function \(H(x, y) = c\) for the given system of differential equations, we are unable to plot level curves as trajectories for this system and indicate the direction of motion on each trajectory. In conclusion, an equation of the form \(H(x, y) = c\) representing the trajectories cannot be derived for the given system of differential equations.

Key concepts

First Integral

The concept of a first integral is essential in the study of differential equations. It refers to a function, typically denoted as \( H(x, y) \), that remains constant along a trajectory of a dynamical system. In other words, for a given system of ordinary differential equations (ODEs), the first integral is a solution that satisfies the system without explicitly involving the independent variable, often time denoted as \( t \).

In the example provided, the intention was to find a first integral for the system \( \frac{dx}{dt} = 2y \) and \( \frac{dy}{dt} = 8x \) by identifying a function \( H(x, y) \) for which its partial derivatives with respect to \( x \) and \( y \) correspond to the right-hand sides of the given ODEs. However, as the solution indicates, an inconsistency arose during the process, which suggests that a first integral of the desired form cannot be found. Understanding why a first integral cannot be found in this scenario is equally important as finding one, as it sheds light on the properties of the system being analyzed.

Level Curves

Level curves, also known as contour lines or isoclines, represent the collection of points where a two-variable function takes on a constant value. Graphically, these curves are indispensable tools for visualizing the topology of a surface described by a function \( H(x, y) \) in a two-dimensional plane.

For the hypothetical function \( H(x, y) \) that we wish to find from the differential equations, level curves would visualize the trajectories of the system on the \(xy\)-plane. Each curve would correspond to a different constant value of \( c \), and plotting a series of these curves would give a fuller picture of the system's behavior. Unfortunately, since we could not obtain an appropriate function \( H(x, y) \), we cannot illustrate the associated level curves and their significance in this specific context.

Partial Derivatives

Partial derivatives are the cornerstones of multivariable calculus. When dealing with functions that depend on two or more variables, like \( H(x, y) \), the partial derivative represents the rate of change of the function with respect to one variable while keeping the other variables held constant. They are notated as \( \frac{\partial}{\partial x} \) or \( \frac{\partial}{\partial y} \) for derivatives with respect to \( x \) and \( y \), respectively.

In the context of our example, the system of ODEs suggests that the partial derivatives of a potential first integral function \( H \) would equate to the expressions given by the system: \( \frac{\partial H}{\partial x} = 2y \) and \( \frac{\partial H}{\partial y} = 8x \). However, as the solution process indicated, deriving a consistent function that fulfills these partial derivatives was not possible. Recognizing the role of partial derivatives in this system highlights the interconnectedness between variables and the complexities that arise when attempting to solve such equations.

Trajectories

In dynamical systems, trajectories represent the paths that a system follows through its phase space as it evolves over time. These paths are solutions to a system of differential equations, and they provide significant insights into the behavior of a system under certain initial conditions.

In solving our example problem, the goal was to find an equation that represented these trajectories explicitly. The trajectories are effectively the paths traced out by solutions \( (x(t), y(t)) \) over time. Typically, a first integral helps us understand these paths by offering conservation laws or invariants. However, due to the inability to determine an appropriate first integral, we could not define these trajectories algebraically in this case. Nonetheless, understanding trajectories is crucial for predicting and controlling the behavior of a wide range of physical and theoretical systems.