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Hello Sun,   Just for your reference (and since I don't have too much time right now), first find the eigenvalues of the matrix.  Do this by solving   (2 - r)(-2 - r) + 5α = 0,   the solutions will be your eigenvalues (r = ±√(4-5α) for your reference)...

Hello Sun,   What's next is simply to solve the quadratic!  So, multiply it out to get   r2 -αr + 5 = 0   and use the quadratic formula to find the roots, which will be your eigenvalues.   Regards, Hassan H.

Hello Sun,   For your reference, since the others haven't settled on an answer, your solution will be:   x = c1( 2, -3, 2 )et + c2et( 0, cos(2t), sin(2t) ) + c3et( 0, sin(2t), -cos(2t) ).   Clearly both methods above/below will work.  Having briefly...

Hello Sun,   Try 1089 X 9 = 9801.   Regards, Hassan H.

Casadi,   You have stumbled upon a series that I used to give my students as a fun challenge!  If you really are stuck, I'll give the answer below, but don't look if you still want to give it a shot!   The first number is 1, so say "one."  Now, to form...

Hello Laura,   In this sentence, the subject is "Blanchard," but the predicate is everything that follows, i.e. "successfully made the first human parachute drop in 1793."  You would put a slash right after "Blanchard" to separate the two.   Blanchard...

Hello Ibukunoluwa,   Let's call the events that the document reaches the destination on time by the first service, A, by the second service, B, and by the third service, C.  Then, the probability that you are interested in computing is the probability that only A or only B or only...

Hello Brandi, Since the Fibonacci sequence is recursively defined by a0 = 1, a1 = 1;  an+2 = an+1 + an for n ≥ 0, we find that the sequence you are interested in, let's call it {bn}, satisfies b1 = a1 - a0 = 0;  bn+2 = an+2 - an+1 = an, for n ≥ 0, by...

Hello Titaree, We can generalize this question a bit, so that you can have a formula for similar problems at hand. Suppose Computer A (or Machine A, Person A, etc.) can do 1 job in x hours, and Computer B can do 1 job in y hours.  These are equivalent to Computer A:    ...

Sun, Proceeding on the assumption that uπ(t) is defined as in my comment above, the Laplace transform is very simple to compute.  Just apply the definition of the transform! £{uπ(t)} = ∫0∞ e-stuπ(t)dt = ∫π∞ e-stdt. I am certain you can compute the above integral. Regards, Hassan...

Hello Sun and Yuan P., While there is nothing wrong with the approach that Yuan takes, he may not be familiar with the line of questions that you have posed recently.  So, I will answer based on the work you have done in the past few questions; namely, you have expressed inverse Laplace...

Hello Sun, The only difference between this problem and the one from yesterday is that you need to know how the Laplace transform behaves when applied to a derivative.  This should be something you have already covered, or at least it should be in your table of transforms, since you are...

Hello Harry, When factoring a quadratic f(x) = ax2+bx+c, you are (usually) seeking integers p and q which satisfy p+q = b pq   = ac simultaneously.  You can check to see whether factoring over the integers is worthwhile by seeing whether the discriminant...

Hello Barbara, The common ratio of a known geometric sequence can be found by r = an+1/an for some valid n.  In your case, of course, the common ratio turns out to be r = 125/625 = 1/5. Now, the formula you presented is not the correct formula for a geometric sequence...

Hello Sun, You will simply want to "split up" the expression for F(s), and then use known (and oftentimes tabulated) results on the Laplace transform to finish the computation.  So, write F(s) = G(s)H(s) where H(s) = 1/(s2+1), and note that the inverse Laplace transform...

Hello Mike, Often, to differentiate a function y = f(x) with a variable base and exponent, you should make use of the identity g(x)h(x) = eh(x) ln g(x) and use your know rules for differentiating the exponential function to proceed.  You will need to use the Product Rule for Differentiation...

Hello Miss., This question from nearly a week ago seems never to have been resolved favorably, so I will attempt to give a brief exposition of its solution.  First, we must be clear on what constitutes a bus ticket.  From the wording of the problem, it seems to me that any string...

Hello Velina, You essentially only have to perform one step in order to arrive at the solution to this equation: x + 7 = 3 x + 7 - 7 = 3 - 7     (Subtract 7 from both sides in order to isolate the x on the left-hand side) x + 0 = -4   (Remember...

Hello Prasenjeet, Briefly, an ion is, as you suspect, a molecule with a net positive or negative charge, while a (free) radical is a molecule (or fragment thereof) with an unpaired electron. Ions are of course ubiquitous in chemistry.  For instance, when we dissolve ammonia gas in water,...

Hello Sun, The Laplace transform of a function f(t) defined for t≥0 is £{f(t)} = ∫0∞ e-st f(t) dt, provided of course that the above converges. In your case, f(t) = cos(at), so you need to evaluate £{cos(at)} = ∫0∞ e-st cos(at) dt. This is an integration-by-parts (twice)...