# Math 2250 Fall 2001
# Project I
# SolutionTemplate
# 
# 1)  Type in your name and student number  here:
# 
#      You are to do the section 1.5 computer project, using this
# template.  When you are done you should print out a copy to hand in to
# your Instructor.  A preliminary discussion, in which the text
# discussion of pages 55-57 is expanded to include Maple commands, is
# found at the end of the tutorial which accompanies this project.  It
# is assumed that you have already worked through that.  Both the
# tutorial and this template can be downloaded by following the links
# from our course home page, at
# http://www.math.utah.edu/~korevaar/2250fall01.html  
# 
# SCENARIO:
#     It is not summer in Georgia.  It is winter in Salt Lake City.  We
# will dream it is early spring, (around equinox), and that the 5-day
# forecast in the newspaper says the weather will be stable, with lows
# of 26 degrees, and highs of 60 degrees.   As in the text and for the
# sake of simplicity, we will assume a sinusoidal daily temperature
# oscillation, except our low will be at 2 a.m, and our  high will be at
# 2 p.m.  
#      We must leave town for 3 days, and are deciding whether to turn
# off the heat while we are gone.  The question we wonder about is, will
# the water pipes in the house freeze if we do turn off the heat?   
# 
# 2)  Review the model which leads to equation (3) on page 56.  Use
# integrals #49 and 50  and the algorithm for solving first order linear
# DE's on pages 44-45, to solve (3) by hand, keeping all parameters as
# letters.   Staple your work onto the printout of your completed
# project which you hand in.  Your answer should agree with equation (4)
# on page 56.  You will see that the constant c0 in that answer is the
# constant C of integration you obtain when you follow the solution
# recipe for linear DE's.  You find its value in terms of the initial
# condition u(0)=u0 by pluggin in u=u0, t=0, as usual.  You will
# discover that the text writes c0 incorrectly:  there is a hidden minus
# sign which has become glued to the fraction which follows a0, in the
# book's formula, i.e. the correct formula for c0 is

                                     2
                     k omega b[1] - k  a[1]
             c[0] := ---------------------- + u[0] - a[0]
                           2        2
                          k  + omega

# 
> restart:with(DEtools):with(plots):
# 3)  Use dsolve to have Maple find the solution to (3), with u(0)=u0. 
# You might have already done this in the tutorial, in which case you
# can copy the appropriate commands  from there, paste them in here, and
# re-execute them.  Check that your solution agrees with your hand work
# above, as well as the (corrected) text.
# 
> 
# 
# 4)  Figure out the parameter values for the Salt Lake City
# temperature, as modeled above.  Of course, omega will still be Pi/12,
# but now you have different daily average temperature, amplitude, and
# phase than was used in equation (2) on page 56, when summer in Georgia
# was being modeled.  Fill the new values in below.  Then work out by
# hand, using the cosine addition formula, the values you should take
# for a0, a1, b1, so that A(t) is given by (1) on page 56, and so that
# the differential equation (3)  also has the correct parameter values. 
# 
# 
# Average Temperature =
# Temperature variation amplitude = 
# Phase delay (was 4 hours in Georgia example) =
# 
> omega:=Pi/12.0;  
> a0:= 
>    #don't forget the semicolons
> a1:=
> b1:=
> A:=t->a0 + a1*cos(omega*t) + b1*sin(omega*t);
# 
# We will assume your house is moderately well-insulated, so that k=0.3:
> k:=0.3;
# 
# 5)  Find the solution to (3) with the particular parameter choices you
# made above.  You should get a solution function like  (5) on page 56,
# but reflecting the Salt Lake City temperatures and the new insulation
# parameter. 
# 
> 
# 6)  Identify the part of your solution which persists as t approaches
# infinity, i.e. the steady periodic solution.  Your formula should have
# the same character as equation (6) on page 57.
# 
> 
# 7)  Write the steady periodic solution in the form of equation (7) on
# page 57, so that you can see the time delay for the inside
# temperature.  You need to use the cosine addition formula.  The text
# has a discussion on page 315 which may help you.  It is shown there
# that if
# 

               x(t) := A cos(omega t) + B sin(omega t)

# then also

                 x(t) := t -> C cos(omega t - alpha)

# where the right triangle of Figure 5.4.4 page 315 summarizes the
# relationships between A,B, C, and alpha.
# 
> 
# 8)  Assume that your heater shut off at midnight, with the inside
# temperature equal to 70 degrees.  Create a plot like Figure 1.5.10
# which displays the inside temperature and the outside ambient
# temperature for the next three days.
# 
> 
# 9)  Create a picture like Figure 1.5.9, which also includes the slope
# field for this differential equation, with our Salt Lake City
# parameters.  Choose initial temperatures between 40 and 70 degrees, in
# 5 degree increments.  Label the maximum and minimum temperature times
# by hand, on the display printout, before you hand in the project.
# 
> 
# 10)  So, based on your work in this project, how likely do you think
# it is that the pipes will freeze if the heater is turned off for 3
# days starting at midnight?  Explain your reasoning.
# 
#