How can I solve this ordinary differential equation?

[mathman61]

Let's say we have a long-range helicopter drone. With this drone, we want to carry a small package to our home. Let's assume that initially the drone was 5 km east of our home. The purpose of the drone will be to fly directly to our house. For this reason, we program the drone to return directly home at a speed of 6 km per hour. Meanwhile, there is a wind blowing from the south at a speed of 4 km per hour. But the drone is always pointing straight to the house. Assuming that the drone always flies at the same height:

1. Obtain the mathematical model that gives the drone's flight path (route) and solve this model using the appropriate analytical method you have learned / will learn in this lesson. (Hint: This model will be a system of ordinary differential equations that can be conveniently reduced to a single ordinary differential equation.)

2. Using the given numerical values (as well as some different values you will give yourself), draw a graph of the drone's journey from the departure location to the home using a computer program.

 

[Romsek]

well the system I get for this is

[MATH]x = \displaystyle \int \limits_0^t ~\dot{x}~d\tau\\ y = \displaystyle \int \limits_0^t ~\dot{y}+4~d\tau\\ \dot{x} = \dfrac{-6x}{\sqrt{x^2+y^2}}\\ \dot{y} = \dfrac{-6y}{\sqrt{x^2+y^2}}[/MATH]
where [MATH]\dot{x},~\dot{y}[/MATH] are the velocities of the drone w/respect to the air.

I leave it to you to figure out how to reduce this to a closed form solution ordinary diff eq.
Mathematica isn't solving it very quickly. It's pretty straightforward to sim though.

I suspect the method is to use polar coordinates since the drone speed is constant. It's will probably get nastily nonlinear though.

 

[HallsofIvy]

Oh! Just as I was typing Romsek posted a very nice solution.

 

[mathman61]

well the system I get for this is

[MATH]x = \displaystyle \int \limits_0^t ~\dot{x}~d\tau\\ y = \displaystyle \int \limits_0^t ~\dot{y}+4~d\tau\\ \dot{x} = \dfrac{-6x}{\sqrt{x^2+y^2}}\\ \dot{y} = \dfrac{-6y}{\sqrt{x^2+y^2}}[/MATH]
where [MATH]\dot{x},~\dot{y}[/MATH] are the velocities of the drone w/respect to the air.

I leave it to you to figure out how to reduce this to a closed form solution ordinary diff eq.
Mathematica isn't solving it very quickly. It's pretty straightforward to sim though.

I suspect the method is to use polar coordinates since the drone speed is constant. It's will probably get nastily nonlinear though.

Thank you Romsek. I tried to solve below like your solving method but I couldn't continue to solve. Can you help me?


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