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

In this problem, we found the function H(x, y) representing the trajectories of the given system of differential equations as \(H(x, y) = -\frac{1}{2}xy + x^2 + C\). After plotting level curves for different values of c, we analyzed the direction of motion on each trajectory using the original system of differential equations.

Step-by-step solution

Finding the function H(x, y)

To find the function \(H(x, y)\) we will look for an integrating factor. Before we start, let's rewrite the given system of differential equations as a vector field and as exact differentials: $$ \begin{pmatrix} dx \\ dy \end{pmatrix} = \begin{pmatrix} 2y \\ -8x \end{pmatrix} dt \Rightarrow dh = 2ydx - 8xdy = 0 $$ Now, let's assume that an integrating factor \(\mu(x, y)\) exists, such that \(\mu \ dh = 0\) is exact. This means: $$ \frac{\partial}{\partial y}(\mu \cdot 2y) = \frac{\partial}{\partial x}(-\mu \cdot 8x) $$ This implies: $$ 2\mu = -8\mu \Rightarrow \mu = -\frac{1}{4} $$ Now, let's multiply our equation for \(dh\) by the integrating factor \(\mu = -\frac{1}{4}\): $$ -\frac{1}{4}(2ydx - 8xdy) = -\frac{1}{2}ydx + 2xdy = dH $$ And finally, we'll integrate to find our function \(H(x, y)\): $$ H(x, y) = \int (-\frac{1}{2}ydx + 2xdy) = -\frac{1}{2}xy + x^2 + C = c $$ Thus, we have the function \(H(x, y) = -\frac{1}{2}xy + x^2 + C\).

Plotting level curves and determining the direction of motion

To plot level curves of our function \(H(x, y)\), we can set \(H(x, y) = c\) for various values of \(c\). This will result in a series of curves on the \(xy\)-plane. Let's consider the following values of \(c\): \(-2\), \(-1\), \(0\), \(1\), \(2\). $$ H(x, y) = -\frac{1}{2}xy + x^2 + C = c $$ For each given value of \(c\), we can solve for \(y\) as a function of \(x\): $$ y = \frac{2(x^2 + C - c)}{x} $$ Now, to determine the direction of motion on each trajectory, we look back at the original system of differential equations: $$ \frac{dx}{dt} = 2y, \quad \frac{dy}{dt} = -8x $$ If \(\frac{dx}{dt} > 0\), then the trajectory is moving to the right, and if \(\frac{dx}{dt} < 0\), it is moving to the left. Similarly, if \(\frac{dy}{dt} > 0\), the trajectory is moving upward, and if \(\frac{dy}{dt} < 0\), it is moving downward. By substituting the expression for \(y\) in terms of \(x\) into the equation for \(\frac{dx}{dt}\), we can compute the direction for each trajectory. Plot the level curves for the selected values of \(c\) and draw arrows on each trajectory to indicate its direction of motion. In conclusion, we found the equation of the form \(H(x, y) = c\) and plotted the corresponding level curves for various values of \(c\). These curves display the trajectories of the given system, along with their respective directions of motion.

Key concepts

Integrating Factor

Understanding the concept of an integrating factor is crucial when dealing with differential equations. An integrating factor is a function, usually denoted as \(\mu(x, y)\), used to multiply a differential equation to make it exact. An exact differential equation is one where a single function's total differential is equivalent to the expression of the differential equation. To find an integrating factor, the goal is to construct a coefficient function that, when multiplied by the non-exact differential, results in an equation that can be integrated directly. In our exercise, by assuming the existence of such a factor, we obtained \(\mu = -\frac{1}{4}\) which made our differential equation exact, leading us to the function \(H(x, y)\).

Level Curves

The graphical representation of functions of two variables, like our \(H(x, y)\), can be effectively visualized using level curves. These curves are created by slicing the graph of the function horizontally at different heights, which correspond to constant values of \(H(x, y)\). As \(c\) varies, we observe different curves on the \(xy\)-plane, each representing a specific value of the function. Such curves help understand the behavior of the function over a range of values and are particularly useful in visualizing the trajectories of the differential equation system. When plotting several level curves for different values of \(c\), we effectively map out the solution landscape for the original system of equations.

Vector Field

A vector field in the context of differential equations represents the rate of change of both variables such that each point in the field is associated with a vector. The vectors indicate the direction and magnitude of the velocity at which a point would move if it followed the trajectory of the system. In our problem, the vector field is constructed from the differential equations \(dx/dt = 2y\) and \(dy/dt = -8x\). These equations define the components of the vectors at each point, visualizing how a point on the plane will travel over time under the influence of the system.

Exact Differentials

Exact differentials arise in the study of functions where the differential can be represented as the sum of partial derivatives with respect to each variable. For a function \(H(x, y)\), the differential \(dH\) is the sum \(H_x dx + H_y dy\), where \(H_x\) and \(H_y\) are the partial derivatives with respect to \(x\) and \(y\), respectively. In our solution, we aimed to find an integrating factor that would transform our differential equation into an exact one, meaning that we could find such a function \(H(x, y)\) whose differential exactly matches the left-hand side of our differential equation.

Trajectory Direction

The direction of a trajectory in a differential equation system tells us how a point moves through the vector field over time. It is defined by the velocity components \(dx/dt\) and \(dy/dt\). Positive \(dx/dt\) means a point moves rightwards, while negative means leftwards. Positive \(dy/dt\) indicates an upwards movement, and negative indicates downwards. In analyzing the trajectory direction, we use the derived function \(H(x, y)\) to substitute values and determine the sign of the velocity components. This helps us predict the behavior of the system and visualize the path that the trajectory will take as it moves across different level curves.