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DIFFERENTIAL EQUATIONS - EXAMPLE 20.2 : During the landing process of an airplane, the velocity is constant at v. (a) If the displacement of the plane is x at time t, find the differential equation that relates t, x and v. (b) The plane has 2 parts of wheels - the front and the back, separated by a distance L. The front part of the wheel touches the land first, that allows the straight body of the plane to form an angle T with the horizontal land. If the vertical distance between the back part of the wheel and the horizontal land is y, find the equation of y as a function of L and T. (c) Find the differential equation that relates dy as a function of dt, v and sin T. (d) Find the differential equation that consist of dy as a function of y, L, v and dt. (e) Find the equation of y as a function of v, L, t and C where C is a constant. (f) When t = 0, prove that y = exp C as the initial value of y.
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Answer Posted / kangchuentat
DIFFERENTIAL EQUATIONS - ANSWER 20.2 : (a) The kinematic relationship is dx = v dt. (b) The trigonometric relationship is sin T = y / L. (c) y = -vt sin T, then dy = -v dt sin T since y / x = sin T and x = -vt for y is decreased with an increasing x. (d) dy = -v dt sin T = -v dt (y / L) = -y (v / L) dt. (e) dy / y = (-v / L) dt. Integrate both sides of equation will produce ln y = -vt / L + C, then y = exp (-vt / L + C) = (exp C) exp (-vt / L). (f) When t = 0, y = (exp C) exp (-vt / L) = (exp C) exp (0) = exp C. The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; chuentat@hotmail.com; http://kangchuentat.wordpress.com.
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