Now
that everyone is working hard on the next assignment, I thought I would
send a few hints. The first four problems are fairly straightforward,
especially I you look over the Maple code I went over in class last week.
Problem 5, the word problem, is naturally a little more difficult. So,
here are some tips now that you have had time to think about the problem.
1.
You need to make sure your units are correct. Keep everything in the
British system. For this you would have the following values for the
constants:
alpha:=0.63; a:=Pi*(3/12)^2; A:=Pi*5^2; g:=32.2; K:=7*231/12^3;
2. If done
correctioly, you should get the differential equation.
diff(h(t), t) = 0.1191451588e-1-0.1263931366e-1*h(t)^(1/2)
3. Maple
cannot handle the exact, algebraic solution. You may have noticed that
using dsolve may give an ugly result. You need to use type=numeric! So, to
get a plot of the height of the water in the take over the two hour range,
you would do the following:
sol:=dsolve({ode2,h(0)=35},type=numeric,range=0..120);
odeplot(sol);
4. In order
to answer the questions, you might want to convert your solution to a
function, or evaluate the solution at various times. For example, typical
expressions might include:
sol(120);
H:=t->rhs(sol(t)[2]);
plot('H(t)',t=0..600);
5. For the
questions to be answered:
a. Flow
rate is given in units of cubic feet per minute. Think about the shape of
the tank as you figure out the change in volume in one minute.
b. Find
the height of the water where the pressure is minimum and do not forget
that the tank is 25 ft above the main pipe into the house.
c. At
what time will the height of the water be 10 feet?
Hopefully, this will help you over any snags.
I
elaborated some hints about the problems in this homework (#6) so that
you might have some direction. I will reiterate, in case you missed what
I said.
Problem 1. Do this like I did in the notes. I did P3 already.
Problem 2. This is straight forward differentiation.
Problem 3. Use the recursion relation and the initial a0 as given. This
will give you only even aj’s. Show that these are the same as what is in
the sum, where j = 2k.
Problem 4. Take the series in Problem 3, multiply it by x^(-n). Now
differentiate the series. Compare this to the series for -x^(-n)
J_(n+1). To do that you need to replace n with n+1 in the series and
compare terms on both sides of the equation.
Problem 5.
b. Start by seeing what you get from Gamma(3/2)=Gamma(1+1/2)=1/2
Gamma(1/2). Now write Gamma(5/2) in terms of Gamma(1/2) .. an continue
until you see a pattern. For induction, 1) show k=0 gives the right
result. 2) Assume the equation true for k and find Gamma(k+1+1/2) in
terms of Gamma(k+1/2). If you replace the expression for Gamma(k+1/2)
will it give the correction expression for k replaced by k+1?
c. You need only replace any appropriate factorials involving n with
the right Gamma function.
Problem
6. Set nu in J_nu of 5 with plus or minus ½. You then factor the right
stuff and should have the series for sine or cosine right there. The
last part is only plugging the given functions into the recursion
formula, like in problem 1.