# Differential Equation of a hawk

• stosw
In summary, the problem involves a hawk and a pigeon, where the hawk flies at a constant speed of 90 ft/sec in the direction of the pigeon, while the pigeon flies at a constant speed of 50 ft/sec in the direction of the y-axis. The hawk's initial position is (3000,0) on the x-axis, while the pigeon's initial position is (0,-2000) on the y-axis. The fact that the hawk is always headed towards the pigeon means that the line PQ is tangent to the pursuit curve y=f(x). To find the equation for the hawk's position, we can use the equation of a line and the coordinates of P and Q. This leads to the equation for h(x,y
stosw

## Homework Statement

Suppose that a hawk, whose initial position is (a,0)=(3000,0) on the x-axis, spots a pigeon at (0,-2000) on the y-axis. Suppose that the pigeon flies at a constant speed of 50 ft/sec in the direction of the y-axis (oblivious to the hawk), while the hawk flies at a constant speed of 90 ft/sec, always in the direction of the pigeon.

The fact that the hawk is always headed in the direction of the pigeon means that the line PQ is tangent to the pursuit curve y=f(x). This tells us that (dy/dx)=h(x,y,t) where h(x,y,t) = ?

## The Attempt at a Solution

I found the equation for the pigeon to be

g(t) = -2000 + 50t

I have no idea how to find an equation for the hawk. Any tips/hints/suggestions would be great.

This question is easier than you might think. Since the hawk is flying towards the pigeon, the tangent line of its path always intersects the pigeon's position. If you write out the equation of the tangent line, it will be obvious what dy/dx is.

stosw said:

## Homework Statement

Suppose that a hawk, whose initial position is (a,0)=(3000,0) on the x-axis, spots a pigeon at (0,-2000) on the y-axis. Suppose that the pigeon flies at a constant speed of 50 ft/sec in the direction of the y-axis (oblivious to the hawk), while the hawk flies at a constant speed of 90 ft/sec, always in the direction of the pigeon.

The fact that the hawk is always headed in the direction of the pigeon means that the line PQ is tangent to the pursuit curve y=f(x). This tells us that (dy/dx)=h(x,y,t) where h(x,y,t) = ?

## The Attempt at a Solution

I found the equation for the pigeon to be

g(t) = -2000 + 50t

I have no idea how to find an equation for the hawk. Any tips/hints/suggestions would be great.
Are you sure you have stated the problem correctly? As it is, as ideasrule said, it is a trivial problem. Perhaps the pigeon is not flying "in the direction of the y-axis" but "parallel to the y-axis"?

Also you said "I found the equation for the pigeon to be g(t) = -2000 + 50t". Well, pigeons don't have equations! If you meant that to be the equation for the position of the pigeon, shouldn't that have both x and y components? Which does that equation give?

i am also confused regarding the question...can anyone post the detailed solution.

HallsofIvy said:
Are you sure you have stated the problem correctly? As it is, as ideasrule said, it is a trivial problem. Perhaps the pigeon is not flying "in the direction of the y-axis" but "parallel to the y-axis"?

Also you said "I found the equation for the pigeon to be g(t) = -2000 + 50t". Well, pigeons don't have equations! If you meant that to be the equation for the position of the pigeon, shouldn't that have both x and y components? Which does that equation give?

I'm almost positive the pigeon is moving only in the direction of the y-axis.

You're right they don't have equations, lol. The pigeon's position Q=(0,g(t)) where i found g(t) = -2000 + 50t.

Could I then say something like the hawk's position P = ( j(t), k(t) ) where

j(t) = 3000 - 90t and k(t) = 90t

(Based on the starting position of the hawk on the x-axis)?

If I can do that, I'm not seeing how to get from here to the equation for h(x,y,t) that involves all three of those variables.

Due to it being the tangent line and me having two points P and Q, could I used y = mx +b? If so, I'm stuck again at how to get to h(x,y,t).

Last edited:

## 1. What is a differential equation of a hawk?

A differential equation of a hawk is a mathematical equation that describes the changes in the population of hawks over time. It takes into account various factors such as birth rate, death rate, and migration to model the population dynamics of hawks.

## 2. Why is it important to study the differential equation of a hawk?

Studying the differential equation of a hawk can provide insights into the population trends of hawks and help in predicting future population sizes. This information is crucial for conservation efforts and managing the impact of hawks on their ecosystems.

## 3. What are the variables in the differential equation of a hawk?

The variables in the differential equation of a hawk may include the population size of hawks, birth rate, death rate, immigration rate, and emigration rate. These variables can vary depending on the specific model being used.

## 4. How is the differential equation of a hawk solved?

The differential equation of a hawk can be solved using various mathematical techniques such as separation of variables, Euler's method, and Runge-Kutta method. The solution will depend on the specific model and initial conditions.

## 5. Can the differential equation of a hawk be applied to other species?

Yes, the differential equation of a hawk can be applied to other species with similar population dynamics. However, the specific variables and parameters may need to be adjusted to fit the characteristics of the species being studied.

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