MATH 285 E1 F1 GRADED HOMEWORK SET 3 DUE WEDNESDAY OCTOBER 8 IN LECTURE This time the homework has just one part Please staple your homework together and put your name and section on it Thank you 1 5 points Show that the functions e3x xe3x and x2 e3x are linearly independent That is show that if c1 e3x c2 xe3x c3 x2 e3x 0 for all x then it necessarily follows that c1 c2 c3 0 Hint Get relations between the ci s by plugging in x 0 and or differentiating the equation 2 15 points Consider the polynomial p r r12 12r11 51r10 r9 1968r8 19003r7 106948r6 440432r5 1423168r4 3448064r3 6069248r2 7606272r 5160960 Find the general solution of the linear homogeneous equation p D y 0 d where as usual D dx and p D denotes the constant coefficient differential operator obtained by substituting D for r in p r We are looking for a general solution that is real not complex and which involves twelve arbitrary constants Hint You should use the following factorization of p r p r r 3 2 r 4 3 r 7 r2 r 5 r2 16 2 3 10 points A body of mass m 5 kg is attached to a spring with spring constant k 20 kg s2 The body is suspended on an airhockey table so that it is not subject to friction or gravity but it is connected to a dashpot mechanism that lets us adjust the degree of damping c 0 The equation of motion is the usual one dx d2 x c kx 0 2 dt dt Consider the following experimental procedure move the body to an initial position x 0 0 m and set it into motion with initial velocity v 0 1 m s Your task is to determine the position function x t in the following three cases a c 10 kg s b c 20 kg s m 1 2 GRADED HOMEWORK 3 c c 30 kg s It is perfectly fine to refer to section 3 4 of the textbook in your solution that is you don t have to rederive everything all over again but you should mention what facts from the textbook you are using 4 5 points Find a particular solution of y 00 4y 0 y 3ex
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