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. $$ d x / d t=-x+y, \quad d y / d t=-x-y $$

Short answer

Answer: The equation satisfied by the trajectories of the given system of differential equations is \(H(x, y) = c = k\frac{-x+y}{-x-y}\). The direction of motion on each trajectory is determined by examining the signs of \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) in the original system of differential equations. The direction of motion can then be indicated by arrows on the level curves plotted for H(x, y).

Step-by-step solution

Rewrite the given system of differential equations

First, let's rewrite the given system of differential equations for better understanding: $$ \frac{dx}{dt} = -x + y \\ \frac{dy}{dt} = -x - y \\ $$

Analyze and find H(x, y)

Now, we will try to find a function \(H(x, y)\) such that the system of differential equations can be expressed in terms of its partial derivatives. To do this, we will analyze the given system and try to find a connection between these two equations: Let's divide the second equation by the first equation: $$ \frac{dy/dt}{dx/dt} = \frac{-x-y}{-x+y} $$ Now, let's rewrite this expression as: $$ \frac{dy}{dx} = \frac{-x-y}{-x+y} $$ This is a first-order differential equation, and we can now try to find a function \(H(x,y)\) that solves it.

Solve the first-order differential equation for H(x, y)

To solve this first-order differential equation, we will use the method of separation of variables: Move all x and y terms to respective sides: $$ \frac{dy}{-x-y} = \frac{dx}{-x+y} $$ Now, let's integrate both sides of the equation: $$ \int \frac{dy}{-x-y} = \int \frac{dx}{-x+y} + C $$ The integral yields: $$ \ln|-x - y| = \ln|-x + y| + C' \\ $$ Where \(C'\) is the constant of integration. Now exponentiate both sides: $$ |-x - y| = k|-x + y| $$ Where k is a new constant. Finally, rewrite this equation as a function of H(x, y): $$ H(x, y) = c = k\frac{-x + y}{-x - y} $$

Plot level curves of H(x, y)

To plot several level curves for the function H(x, y), we can create different values for c and assign them to the equations, obtaining different trajectories for the system of equations. For example, if \(c = 1, 2, 3, ...,\) we will get various solutions for the function H(x, y). The level curves of the function H will act as trajectories, and we can plot these curves to visualize the behavior of the system.

Indicate the direction of motion on each trajectory

To indicate the direction of motion, we have to analyze the given system of differential equations. By examining the signs of the equations \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\), we can determine the overall direction of motion for each trajectory. For example, if \(\frac{dx}{dt} > 0\) and \(\frac{dy}{dt} > 0\), the direction of motion on that particular trajectory will be towards the right and upwards. Similarly, other combinations of signs for \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) will determine the other directions of motion. These directions of motion can be indicated by arrows on the level curves represented in the plot.

Key concepts

Level curves of function H

Let's delve into the fascinating concept of level curves of a function, specifically for a function denoted as H(x, y). These curves are significant in visualizing the solutions to certain types of differential equations. Imagine you are hiking up a hill, and a map shows you lines that represent points of equal altitude; these are level curves. In mathematics, for a given function H(x, y), level curves are the collection of points that satisfy the equation H(x, y) = c, where c is a constant.

These curves help us understand the behavior of a system at different levels of H, and each curve corresponding to a particular value of c is a trajectory of the system. This means that along each curve, the function H does not change – it's constant, just like how the altitude of a contour line on a map doesn't change. Visualizing these level curves gives us insight into the system's behavior without solving the entire differential equation at every point.

First-order differential equation

A first-order differential equation is a relationship involving rates of change and is fundamental in various fields like physics, engineering, and economics. For our purposes, it's an equation that contains first derivatives — and no higher — of an unknown function. Given the equation \(\frac{dy}{dx} = f(x, y)\), where f is some function of x and y, our goal is to find the unknown function y that satisfies this relationship.

The equations given in our problem, \(dx/dt = -x + y\) and \(dy/dt = -x - y\), are excellent examples of first-order differential equations, as they relate the rates of change of x and y with respect to time t. They illustrate how each variable impacts the system's trajectory over time.

Method of separation of variables

One of the most intuitive techniques for solving first-order differential equations is the method of separation of variables. It involves rearranging the equation such that all terms involving x are on one side, and all terms involving y are on the other side, which then allows us to integrate both sides separately.

Consider the equation \(\frac{dy}{dx} = g(x)h(y)\) as a basic form for separating variables. The idea is to isolate \(dy/h(y)\) and \(g(x)dx\) on opposite sides, leading to \(\int\frac{dy}{h(y)} = \int g(x)dx\). This process converts our differential equation into a problem of finding anti-derivatives, making it simpler to solve. In our exercise, after separating the variables, we get \(\int\frac{dy}{-x-y} = \int\frac{dx}{-x+y}\), setting the stage for integration and bringing us closer to finding function H.

Direction of motion on trajectory

Understanding the direction of motion on a trajectory is crucial when analyzing dynamical systems. In our context, a trajectory is the path that the system follows as time progresses. To determine the direction of motion, we look at the signs and magnitudes of the derivatives \(dx/dt\) and \(dy/dt\). These give us the velocity components in both the x and y directions.

If both \(dx/dt\) and \(dy/dt\) are positive, the motion is towards the top right of the phase plane. Similarly, if \(dx/dt\) is positive but \(dy/dt\) is negative, the motion is towards the bottom right. This concept is particularly useful when we want to sketch the trajectories on a phase plane. By drawing arrows along the level curves of H(x, y), we provide visual cues of how a point would move over time if it started on that curve.