Question

Problem 7 If A cos 0,+B sin oft = rsin(0 -0) = R cos(t-8), (a) determiner and in terms of A and B , and (b) the relationship among R,r, 8 and 2 Ans. (a) r = VA’ + Bº, tan 0 4 b)r=R, 0 = 8-5, tan tan 8 +1 = 0 Problem 8 (a) Define the steady state solution of a given SLDE. (b) Consider the motion of a 2-kg mass in a (m,c,k)-system under a cyclic load. The governing differential equation is mü + củ + ku = 5 cos where m = 2kg , c = 1 N. and k = 3 Determine the amplitude and phase shift of the steady state response of the mass. Ans: Amplitude = 0.327, Phase = -0.721 111 Problem 9 (a) Using the method of reduction of order, derive the second fundamental solution of the homogeneous SLDE x’y" - 3xy' +4y = 0, y(x) = x². (b) Write down the general solution. Problem 10 Using the method of variation of parameters, show that a particular solution of y" –2y – 3y = xe**, is given by y (x) = - * [8x²+4x+1] If y(O)=1, y'(0)=0, derive the complete solution of the nonhomogeneous SLDE. Ans: y(x) = 0.266e3x +0.75e-* - (8x2 + 4x + 1)

Answer 1

and tauo- taus = -A 2) tano tame = -1 » tano taus +1=0/