Differential equation word problem

In summary: I think you should change (0,0) to (10,0) in the equation. You can also use the equation to find dx/dt and dy/dt in terms of x,y,Sa,and Sw. Any help would be helpful. Thanks
  • #1
bobbyz
3
0
The problem starts by saying that an aircraft is flying such that it's longitudnal axis always points towards the point (0,0). The plane itself starts at some point on the x-axis (say like (10,0)). The plane always has a constant airspeed, and experiences a constant wind blowing "north"

We want to find the path the aircraft takes

My solution so far:

Let s1=speed of the plane
s2=speed of the wind
theta=angle of the plane at time t

Find sin(theta) and cos(theta) in terms of x and y:
y=r sin(theta) and x=r cos(theta)
I think this is right if I remember my calc correctly, but I am not sure if r can be written using the speeds s1 and s2?

The next part of the question asks us to find dx/dt and dy/dt in terms of x,y,s1, and s2:

I am not sure how to do this part. I think I should break up the planes velocity into its components parallel to the axis but that is as far as I have gotten. If I could have help here I think I could finish the rest of the problem, but I don't want to mess up my initial work and do the whole problem wrong.

Thanks
 
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  • #2
Since the airplane is always "pointed toward" (10, 0), if it is at (x,y) then the tangent line to its path is the line from (10,0) to (x,y) which has slope y/(x-10). That means that, along its path dy/dx= y/(x-10), a simple separable differential equation. You can solve that for y as a function of x. Let its velocity vector with respect to the air by <u, v> and the wind's velocity be <0,w>. Then its velocity vector with respect to the ground is <u, v+ w> so dx/dt= u, dy/dt= v+ w. While w is a constant, u and v are not but the fact that the airplane's airspeed is a constant means that [itex]\sqrt{u^2+ w^2}= C[/itex] is a constant. You should be able to put that together with y as a function of x to find dx/dt and dy/dt in terms of the constants w and C.
 
  • #3
I think you made some errors HallsofIvy.

The plane is always pointed towards (0,0). This would mean that the tangent line slope is y/x. Am I right?
 
  • #4
Also, if anyone reads this, I am still having trouble finding dx/dt and dy/dt.

I have found that the total x direction motion over time can be found by x=Sa(x/sqrt(x^2+y^2)) and the total y motion over time can be found by y=Sa(y/sqrt(x^2+y^2)) +Sw. I am pretty sure that this is right but my question is how to dx/dt and dy/dt in terms of x,y,Sa,and Sw. Any help would be helpful. Thanks
 
  • #5
bobbyz said:
I think you made some errors HallsofIvy.

The plane is always pointed towards (0,0). This would mean that the tangent line slope is y/x. Am I right?
Yes, I got (0,0) and (10,0) mixed up!
 

What is a differential equation word problem?

A differential equation word problem is a mathematical problem that involves finding a function or equation that describes a relationship between a variable and its rate of change. It can involve real-world situations, such as growth and decay of populations, motion of objects, or chemical reactions.

What are the steps to solve a differential equation word problem?

The steps to solve a differential equation word problem are:

  1. Identify the dependent and independent variables.
  2. Write out the given information and any known conditions.
  3. Use the given information and conditions to write a differential equation.
  4. Solve the differential equation using integration techniques.
  5. Check the solution by plugging it back into the original equation.

What are some common types of differential equation word problems?

Some common types of differential equation word problems include:

  • Population growth or decay problems
  • Motion or velocity problems
  • Chemical reaction rate problems
  • Circuit analysis problems
  • Heat transfer problems

How do differential equation word problems relate to real-world situations?

Differential equation word problems are used to model and solve real-world situations involving rates of change. They can help predict and understand how a system will behave over time, such as the growth or decay of a population, the movement of an object, or the reaction of chemicals.

What are some strategies for approaching a difficult differential equation word problem?

Some strategies for approaching a difficult differential equation word problem include:

  • Identify the type of differential equation and use appropriate techniques to solve it.
  • Break the problem into smaller parts and solve each part separately.
  • Use known conditions and information to set up the differential equation.
  • Consult with peers or a teacher for help and guidance.

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