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Differential Equations

October 31, 2015 0 Comment

QuestionAn aircraft flying under the guidance of a nondirectional beacon (NDB),
which is a fixed radio transmitter, moves so that its longitudinal axis always points in the direction of the beacon. Initially the aircraft is heading towards the beacon with constant velocity or airspeed “v” from a position with the wind at right angles to this initial direction. The wind does not change directions and its velocity is a constant “w”.

a) If the trip starts at (2,0) and ends at (0,0), set up the ordinary differential equation. describing the aircraft’s path over the ground given that

dy/dx = (dy/dt)/(dx/dt)

b) Make the substitution v = y/x to solve this equation.

c) Use the initial values for x and y to determine the value of the arbitrary
constant found in part b), and solve to get y explicitly in terms of x.

d) Discuss the cases the ratio of windspeed to airspeed = 1 and > 1. What will happen to the aircraft?

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