Chasing Ducks: Solving a Differential Equation in a Square Pen

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In summary, the problem involves four ducks situated in the corners of a 2mx2m square pen and each duck begins to chase its anticlockwise neighbour at the same speed. The question asks to find the differential equation governing the path of the duck starting at the lower left corner, with the initial condition also being asked. The solution involves finding expressions for dy/dt and dx/dt and then using dy/dt=(dy/dx)*(dx/dt) to find dy/dx. However, the problem is actually a drawing problem and the first duck's position can be determined using the home duck's position.
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Anabelle37
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Homework Statement



one duck is situated in each of the four corners of a 2mx2m square pen. suddenly, each duck begins to chase its anticlockwise neighbour. all the ducks travel at the same speed.

(a) by defining the origin of an x-y coordinate system in the bottom left hand corner of the paddock, show that the differential equation governing the path of the duck is dy/dx = (x-y)/(2-x-y); where (x,y) represents the position of the duck starting in the lower left corner.
what is the initial condition for the position of this duck.

The Attempt at a Solution



I have attached the file which shows the ducks in the four corners. I've also added a circle inside the square pen which represents the path of the ducks (not sure if this is right). I don't know where to go from here. I know I have to find expressions for dy/dt and dx/dt then use dy/dt=(dy/dx)*(dx/dt) to find dy/dx but I don't know how to find dx/dt and dy/dt?
Please help. It's revision for an exam!
Thanks
 

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  • #2
the mighty ducks …

Hi Anabelle! :smile:

hmm … this is a drawing problem, rather than a calculus one

hint: if home duck is at (x,y), where is first duck? :wink:
 

1. What is a pursuit problem in ODE?

A pursuit problem in ODE (ordinary differential equations) is a type of mathematical problem that involves finding a solution for the motion of a moving object that is trying to catch up to another moving object. It is commonly used in fields such as physics, engineering, and biology to model real-world scenarios.

2. How is a pursuit problem solved using ODE?

A pursuit problem in ODE is typically solved by setting up a system of differential equations that describes the motion of both the pursuer and the target. This system is then solved using numerical or analytical methods, such as Euler's method or the Runge-Kutta method. The solution provides the position and velocity of the pursuer at any given time, allowing them to catch up to the target.

3. What are some applications of pursuit problems?

Pursuit problems have many applications in various fields. For example, they can be used to model predator-prey relationships in ecology, missile guidance systems in engineering, and pursuit and evasion tactics in warfare and sports. Pursuit problems can also be used to study optimal control and optimization problems.

4. What are the limitations of using ODE to solve pursuit problems?

While ODE is a powerful tool for solving pursuit problems, it does have some limitations. One of the main limitations is that it assumes a perfect and continuous motion of both the pursuer and the target, which may not always be the case in real-world scenarios. Additionally, ODE does not take into account external factors such as obstacles or changes in the environment, which may affect the motion of the objects.

5. How are pursuit problems related to other mathematical concepts?

Pursuit problems are closely related to other mathematical concepts such as optimal control theory, differential games, and the calculus of variations. These concepts all involve finding the best possible solution for a system that is dependent on time and involves multiple variables. Additionally, pursuit problems can also be solved using other mathematical methods such as linear algebra and optimization techniques.

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