O.k.: that's a nasty and rather realistic problem. Very often "real" problems aren't just initial value problems, they are "mixed" boundary value problems. A flexing beam (as mech. engineers should know or will learn) was analyzed by Euler several hundred years ago, and the analysis constructed makes a model which is a FOURTH ORDER o.d.e., and, depending how the beam is fixed or can move at both ends, leads to problems similar to this one.
So I just solved it. Here is what I did. To begin with, I ignored the condition y(L)=0. I don't know a simple Laplace transform technique which would enable me to build in a boundary condition at x=L.
I considered the problem
y(4)(x)=(Const)delta(x-a) where (Const) was the quotient M/EI.
I also used the initial conditions
y(0)=0 and y''(0)=0 and y(3)(0)=F_0 (given by the textbook statement) AND the initial condition y'(0)=B where B is a constant I will figure out later. I guessed that if I leave one parameter free in the problem statement, that, with luck, I will be able to deduce a value of B uniquely which will correspond to y(L)=0. This is very SNEAKY.
Then I took the Laplace transform of both sides of the equation. Then
I imposed the initial conditions (including the "fictitious" one). I
also assumed that 0
I solved for Y(s), and transformed back. The expressions involving s
were not difficult to inverse Laplace transform. I then plugged in x=L
and tried to get 0 by specifying B correctly, and I was able to do
it. (Sometimes this is NOT possible -- I will try to mention this in
class tomorrow.) I think I got something which looks very much like
what is in the back of the book. All the darn letters make it much
harder to write.
2/9/2004: Sent in response to a request for help
doing problem #19 in section 3.5
2/9/2004: Sent in response to a request for help doing problem #19 in section 3.5
There is some "physical" information in the problem statement. The first sentence ("8 pounds ... 6 inches") gives information about the spring constant.
Then the problem "kicks" the spring with a delta function. if you model it with delta(t-1) you would look at the velocity of the solution at t=1. If you try delta(t) (which I think I would do) then you need the velocity at t=0. Of course, the two numbers should be the same. Consideration of the period and magnitude corresponds.
Look at the function exp(-Lx^3). I will use L for lambda. Remember L is positive and x is in the interval [2,14].
Observe (in your head!) these features of the graph:
Another student asked about problem #7
This is what Maple tells me is the antiderivative of x(cos(5x))7:
/ 6 4 1/5 x |1/7 cos(5 x) sin(5 x) + 6/35 cos(5 x) sin(5 x) \ 2 16 \ 7 + 8/35 cos(5 x) sin(5 x) + -- sin(5 x)| + 1/1225 cos(5 x) 35 / 5 3 16 + 6/4375 cos(5 x) + 8/2625 cos(5 x) + --- cos(5 x) 875You certainly don't want to verify or use this. Please instead use symmetry and just try to answer the question (are the integrals 0 or positive or negative). You don't need to evaluate the integrals!