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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Diﬀerential Equations Week 05 Worksheet (February 28, 2005): Second-Order Linear Equations Spring 2004-05 Q1. (The method of variation of parameters) The homogeneous equation ty − (t + 1) y + y = 0 (t > 0) has two linearly independent solutions y1 (t) = t + 1 and y2 (t) = et . By the method of variation of parameters, the nonhomogeneous equation ty − (t + 1) y + y = t2 , t>0 has solutions of the form u1 (t) · (t + 1) + u2 (t) · et for suitable choices of functions u1 (t) and u2 (t). (a) Assuming that u1 (t) · (t +1)+ u2 (t) · et = 0, derive the additional condition (another equation) that u1 (t) and u2 (t) must satisfy for u1 (t) · (t + 1) + u2 (t) · et to be a solution of the nonhomogeneous equation. (b) From the two equations on u1 (t) and u2 (t) in (a), ﬁnd the unknown functions u1 (t) and u2 (t), and hence the general solution of the nonhomogeneous equation. Q2. (Mechanical vibrations) The vibration of a single-wheel bike travelling along a bumpy road can be modeled by the diﬀerential equation my + γy + ky = f (t). Suppose the mass m = 200 kg, the spring constant k = 10000 N/m, and the bumpy road is modeled by a cosine function of amplitude 1 meter and period 16 meters so that the external force acting on the 2πvt bike travelling with constant speed v (in m/s) is f (t) = 200 cos . 16 (a) Suppose the shock absorber has a damping constant γ = 3000 Ns/m, and the speed of the bike is v = 4 m/s. (i) Write down the equation of vibration for the bike and ﬁnd its general solution. (ii) Is it possible that the vibration will eventually die down as t → ∞? Why? (b) Suppose the shock absorber of the bike has been worn out and provides no damping any more (i.e., γ = 0). At what travelling speed v will resonance occur? ...
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