The Laplace transform, Y(s) =L{y(t)} , can be used to solve the linear differential
equations with variable monomial coefficients, 2y´´+ ty´− 2y = 10 , y(0) = y´(0) = 0 .
Please solve Y(s) , and then find y(t) =L−1Y(s)} ,
y(t) = 5t2
After a mass m is attached to an ‘aging’ spring, K(t) = ke−αt , the free un-damped
motion equation can be transform into a different equation of
P(s)x´´ +Q(s)x´ + s2x = 0 with the change of variables, s = f(t) . Find x(t) .
P(s) = s2
Q(s) = 1
x(t) = c1sin(t) + c2cos(t)
Q(s) = s
Li-Chi is a college student who knows probability very well. He claims that (1) the
probability of any event must be positive, and (2) the sum of the probabilities of all
events for any given experiment must be 1. Do you agree with his claims? If yes,
please simply write “YES” in your answer sheet. Otherwise, please correct his claim(s)
(Your answer must be less than 20 words in total in English or Chinese or you will
receive ZERO credit)
Li-Chi throws a fair six-sided die continuously. How many 5's will he observe on
average before finally getting two 2's?
Let X and Y denote the time when the first and second lightings strike in a stormy
night. Note that X and Y are continuous random variables with a joint PDF
,and the condition PDF fX|Y(x|y).(Please specify the range of U, x,
and y in your answers or you will receive ZERO credit.)
Li-Chi is playing a video game shown below. His goal is to destroy the space ship
using the machine gun. Assume that the ship moves horizontally at a speed of 8
cm/second and it bounces back immediately when reaching each side. The ship starts
at the left side at t = 0 . Assume that Li-Chi fires once randomly within each second
and the “bullet” flies at an infinitely high speed. What is the probability that he can hit
the space ship for every firing? What is the average time (in seconds) it takes to
hit the ship?
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